data_source
stringclasses 9
values | question_number
stringlengths 14
17
| problem
stringlengths 14
1.32k
| answer
stringlengths 1
993
| license
stringclasses 8
values | data_topic
stringclasses 7
values | solution
stringlengths 1
991
|
|---|---|---|---|---|---|---|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.13
|
Calculate $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})$ and $\mathbf{u} \times(\mathbf{v} \times \mathbf{w})$:
$\mathbf{u}=(1,1,1), \mathbf{v}=(3,0,2), \mathbf{w}=(2,2,2)$
|
0 and $(8,-10,2)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0 and $(8,-10,2)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.7.1
|
Let $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}$ in Cartesian coordinates. Find the Laplacian of $f$ in spherical coordinates.
|
$12 \rho $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
12 \rho
|
college_math.Corrals_Vector_Calculus
|
exercise.1.6.3
|
Determine if the given equation describes a sphere. If so, find its radius and center: $2 x^{2}+2 y^{2}+2 z^{2}+4 x+4 y+4 z-44=0$
|
radius: 5, center: $(-1,-1,-1)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
radius: 5, center: $(-1,-1,-1)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.9
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$
|
does not exist
|
GNU Free Documentation License
|
college_math.vector_calculus
|
does not exist
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.15
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{y^{4} \sin (x y)}{x^{2}+y^{2}}$
|
$0 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0
|
college_math.Corrals_Vector_Calculus
|
exercise.3.3.5
|
Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x+2 y$, $P=(2,1,4)$.
|
$x+2 y=z $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
x+2 y=z
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.1
|
Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z ; \quad C: x=\cos t, y=\sin t, z=t, 0 \leq t \leq 2 \pi$
|
$2 \sqrt{2} \pi^{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 \sqrt{2} \pi^{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.3.3
|
Find the angle $\theta$ between the vectors $\mathbf{v}=(5,1,-2)$ and $\mathbf{w}=(4,-4,3)$.
|
$73.4^{\circ} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
73.4^{\circ}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.15
|
Calculate $(\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{z})$:
$\mathbf{u}=(1,1,1), \mathbf{v}=(3,0,2), \mathbf{w}=(2,2,2), \mathbf{z}=(2,1,4)$
|
$14 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
14
|
college_math.Corrals_Vector_Calculus
|
exercise.4.7.1
|
Evaluate the integral
$$
\int_{-\infty}^{\infty} e^{-x^{2}} d x
$$
using anything you have learned so far.
|
$\sqrt{\pi}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\sqrt{\pi}
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.3
|
How many seven-element subsets are there in a set of nine elements?
|
$\left(\begin{array}{l}9 \\ 7\end{array}\right)=36$
|
GNU Free Documentation License
|
college_math.probability
|
\left(\begin{array}{l}9 \\ 7\end{array}\right)=36
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.2.9
|
A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome $X$. Find $V(X)$ and $D(X)$.
|
$V(X)=\frac{20}{9}, \quad D(X)=\frac{2 \sqrt{5}}{3}$.
|
GNU Free Documentation License
|
college_math.probability
|
V(X)=\frac{20}{9}, \quad D(X)=\frac{2 \sqrt{5}}{3}$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.21
|
Modify the program AllPermutations to count the number of permutations of $n$ objects that have exactly $j$ fixed points for $j=0,1,2, \ldots, n$. Run your program for $n=2$ to 6 . Make a conjecture for the relation between the number that have 0 fixed points and the number that have exactly 1 fixed point. A proof of the correct conjecture can be found in Wilf. ${ }^{12}$
|
They are the same.
|
GNU Free Documentation License
|
college_math.probability
|
They are the same.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.2.7
|
A coin is tossed three times. Let $X$ be the number of heads that turn up. Find $V(X)$ and $D(X)$.
|
$V(X)=\frac{3}{4}, \quad D(X)=\frac{\sqrt{3}}{2}$.
|
GNU Free Documentation License
|
college_math.probability
|
V(X)=\frac{3}{4}, \quad D(X)=\frac{\sqrt{3}}{2}$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.3.3
|
The lifetime, measure in hours, of the ACME super light bulb is a random variable $T$ with density function $f_{T}(t)=\lambda^{2} t e^{-\lambda t}$, where $\lambda=.05$. What is the expected lifetime of this light bulb? What is its variance?
|
$\mu=40, \sigma^{2}=800$
|
GNU Free Documentation License
|
college_math.probability
|
\mu=40, \sigma^{2}=800
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.1.5
|
In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.
|
$-1 / 19$
|
GNU Free Documentation License
|
college_math.probability
|
-1 / 19
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.5
|
Use the program BinomialProbabilities to find the probability that, in 100 tosses of a fair coin, the number of heads that turns up lies between 35 and 65 , between 40 and 60 , and between 45 and 55 .
|
.998,.965,.729
|
GNU Free Documentation License
|
college_math.probability
|
.998,.965,.729
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.1
|
Four people are to be arranged in a row to have their picture taken. In how many ways can this be done?
|
24
|
GNU Free Documentation License
|
college_math.probability
|
24
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.5.1.27
|
Assume that the probability that there is a significant accident in a nuclear power plant during one year's time is .001. If a country has 100 nuclear plants, estimate the probability that there is at least one such accident during a given year.
|
$m=100 \times(.001)=.1$. Thus $P$ (at least one accident $)=1-e^{-.1}=.0952$.
|
GNU Free Documentation License
|
college_math.probability
|
m=100 \times(.001)=.1$. Thus $P$ (at least one accident $)=1-e^{-.1}=.0952$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.1.13
|
You have 80 dollars and play the following game. An urn contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune?
|
45
|
GNU Free Documentation License
|
college_math.probability
|
45
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.1.1
|
A card is drawn at random from a deck consisting of cards numbered 2 through 10. A player wins 1 dollar if the number on the card is odd and loses 1 dollar if the number if even. What is the expected value of his winnings?
|
$-1 / 9$
|
GNU Free Documentation License
|
college_math.probability
|
-1 / 9
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.2.2.15
|
At the Tunbridge World's Fair, a coin toss game works as follows. Quarters are tossed onto a checkerboard. The management keeps all the quarters, but for each quarter landing entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of a quarter, and that the outcomes are described by coordinates chosen at random. Is this a fair game?
|
Yes.
|
GNU Free Documentation License
|
college_math.probability
|
Yes.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.15
|
A computing center has 3 processors that receive $n$ jobs, with the jobs assigned to the processors purely at random so that all of the $3^{n}$ possible assignments are equally likely. Find the probability that exactly one processor has no jobs.
|
$\frac{\left(\begin{array}{l}3 \\ 1\end{array}\right) \times\left(2^{n}-2\right)}{3^{n}}$.
|
GNU Free Documentation License
|
college_math.probability
|
\frac{\left(\begin{array}{l}3 \\ 1\end{array}\right) \times\left(2^{n}-2\right)}{3^{n}}$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.4.1.19
|
In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9 , but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does?
|
.273.
|
GNU Free Documentation License
|
college_math.probability
|
.273.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.11.2.9
|
A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State five is an absorbing state. Find the expected number of steps to reach state five.
|
2.08
|
GNU Free Documentation License
|
college_math.probability
|
2.08
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.5.1.17
|
The probability of a royal flush in a poker hand is $p=1 / 649,740$. How large must $n$ be to render the probability of having no royal flush in $n$ hands smaller than $1 / e$ ?
|
649741
|
GNU Free Documentation License
|
college_math.probability
|
649741
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.4.1.21
|
It is desired to find the probability that in a bridge deal each player receives an ace. A student argues as follows. It does not matter where the first ace goes. The second ace must go to one of the other three players and this occurs with probability $3 / 4$. Then the next must go to one of two, an event of probability $1 / 2$, and finally the last ace must go to the player who does not have an ace. This occurs with probability $1 / 4$. The probability that all these events occur is the product $(3 / 4)(1 / 2)(1 / 4)=3 / 32$. Is this argument correct?
|
No.
|
GNU Free Documentation License
|
college_math.probability
|
No.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.4.3.1
|
One of the first conditional probability paradoxes was provided by Bertrand. ${ }^{23}$ It is called the Box Paradox. A cabinet has three drawers. In the first drawer there are two gold balls, in the second drawer there are two silver balls, and in the third drawer there is one silver and one gold ball. A drawer is picked at random and a ball chosen at random from the two balls in the drawer. Given that a gold ball was drawn, what is the probability that the drawer with the two gold balls was chosen?
|
$2 / 3$
|
GNU Free Documentation License
|
college_math.probability
|
2 / 3
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.21
|
A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?
|
$3\left(2^{5}\right)-3=93$ (we subtract 3 because the three pure colors are each counted twice).
|
GNU Free Documentation License
|
college_math.probability
|
3\left(2^{5}\right)-3=93$ (we subtract 3 because the three pure colors are each counted twice).
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.8.1.5
|
Let $X$ be a random variable with $E(X)=0$ and $V(X)=1$. What integer value $k$ will assure us that $P(|X| \geq k) \leq .01$ ?
|
$k=10$
|
GNU Free Documentation License
|
college_math.probability
|
k=10
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.2.1
|
A number is chosen at random from the set $S=\{-1,0,1\}$. Let $X$ be the number chosen. Find the expected value, variance, and standard deviation of $X$.
|
$E(X)=0, V(X)=\frac{2}{3}, \quad \sigma=D(X)=\sqrt{\frac{2}{3}}$.
|
GNU Free Documentation License
|
college_math.probability
|
E(X)=0, V(X)=\frac{2}{3}, \quad \sigma=D(X)=\sqrt{\frac{2}{3}}$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.5.2.37
|
Let $X$ be a random variable having a normal density and consider the random variable $Y=e^{X}$. Then $Y$ has a $\log$ normal density. Find this density of $Y$.
|
$F_{Y}(y)=\frac{1}{\sqrt{2 \pi y}} e^{-\frac{\log ^{2}(y)}{2}}$, for $y>0$.
|
GNU Free Documentation License
|
college_math.probability
|
F_{Y}(y)=\frac{1}{\sqrt{2 \pi y}} e^{-\frac{\log ^{2}(y)}{2}}$, for $y>0$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.1.2.9
|
A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability $5 / 8$, French with probability $5 / 8$, and art and French together with probability $1 / 4$. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?
|
$3 / 4,1$
|
GNU Free Documentation License
|
college_math.probability
|
3 / 4,1
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.8.1.1
|
A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is $(100 \cdot 1 / 2 \cdot 1 / 2)^{1 / 2}=5$. What does Chebyshev's Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15 )?
|
$1 / 9$
|
GNU Free Documentation License
|
college_math.probability
|
1 / 9
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.9
|
Find integers $n$ and $r$ such that the following equation is true:
$$
\left(\begin{array}{c}
13 \\
5
\end{array}\right)+2\left(\begin{array}{c}
13 \\
6
\end{array}\right)+\left(\begin{array}{c}
13 \\
7
\end{array}\right)=\left(\begin{array}{l}
n \\
r
\end{array}\right)
$$
|
$n=15, r=7$
|
GNU Free Documentation License
|
college_math.probability
|
n=15, r=7
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.4.1.13
|
Two cards are drawn from a bridge deck. What is the probability that the second card drawn is red?
|
$1 / 2$
|
GNU Free Documentation License
|
college_math.probability
|
1 / 2
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.11.1.1
|
It is raining in the Land of Oz. Determine a tree and a tree measure for the next three days' weather. Find $\mathbf{w}^{(1)}, \mathbf{w}^{(2)}$, and $\mathbf{w}^{(3)}$ and compare with the results obtained from $\mathbf{P}, \mathbf{P}^{2}$, and $\mathbf{P}^{3}$.
|
$\mathbf{w}(1)=(.5, .25, .25)$
$\mathbf{w}(2)=(.4375, .1875, .375)$
$\mathbf{w}(3)=(.40625, .203125, .390625)$
|
GNU Free Documentation License
|
college_math.probability
|
\mathbf{w}(1)=(.5, .25, .25)$
$\mathbf{w}(2)=(.4375, .1875, .375)$
$\mathbf{w}(3)=(.40625, .203125, .390625)
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.7.1.3
|
Let $X_{1}$ and $X_{2}$ be independent random variables with common distribution
$$
p_{X}=\left(\begin{array}{ccc}
0 & 1 & 2 \\
1 / 8 & 3 / 8 & 1 / 2
\end{array}\right) .
$$
Find the distribution of the sum $X_{1}+X_{2}$.
|
$\quad\left(\begin{array}{ccccc}0 & 1 & 2 & 3 & 4 \\ \frac{1}{64} & \frac{3}{32} & \frac{17}{64} & \frac{3}{8} & \frac{1}{4}\end{array}\right)$
|
GNU Free Documentation License
|
college_math.probability
|
\quad\left(\begin{array}{ccccc}0 & 1 & 2 & 3 & 4 \\ \frac{1}{64} & \frac{3}{32} & \frac{17}{64} & \frac{3}{8} & \frac{1}{4}\end{array}\right)
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.6.1.3
|
In a class there are 20 students: 3 are 5' 6”, 5 are 5'8”, 4 are 5'10", 4 are 6 ', and 4 are 6' 2". A student is chosen at random. What is the student's expected height?
|
$5^{\prime} 10.1^{\prime \prime}$
|
GNU Free Documentation License
|
college_math.probability
|
5^{\prime} 10.1^{\prime \prime}
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.5.1.13
|
The Poisson distribution with parameter $\lambda=.3$ has been assigned for the outcome of an experiment. Let $X$ be the outcome function. Find $P(X=0)$, $P(X=1)$, and $P(X>1)$.
|
.7408,.2222, .0370
|
GNU Free Documentation License
|
college_math.probability
|
.7408,.2222, .0370
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.4.1.43
|
The Yankees are playing the Dodgers in a world series. The Yankees win each game with probability .6. What is the probability that the Yankees win the series? (The series is won by the first team to win four games.)
|
.710.
|
GNU Free Documentation License
|
college_math.probability
|
.710.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.3
|
In a digital computer, a bit is one of the integers $\{0,1\}$, and a word is any string of 32 bits. How many different words are possible?
|
$2^{32}$
|
GNU Free Documentation License
|
college_math.probability
|
2^{32}
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.7
|
Five people get on an elevator that stops at five floors. Assuming that each has an equal probability of going to any one floor, find the probability that they all get off at different floors.
|
$\frac{5 !}{5^{5}}$.
|
GNU Free Documentation License
|
college_math.probability
|
\frac{5 !}{5^{5}}$.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.11
|
A restaurant offers apple and blueberry pies and stocks an equal number of each kind of pie. Each day ten customers request pie. They choose, with equal probabilities, one of the two kinds of pie. How many pieces of each kind of pie should the owner provide so that the probability is about .95 that each customer gets the pie of his or her own choice?
|
Eight pieces of each kind of pie.
|
GNU Free Documentation License
|
college_math.probability
|
Eight pieces of each kind of pie.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.15
|
A baseball player, Smith, has a batting average of .300 and in a typical game comes to bat three times. Assume that Smith's hits in a game can be considered to be a Bernoulli trials process with probability .3 for success. Find the probability that Smith gets $0,1,2$, and 3 hits.
|
.343,.441, .189, .027.
|
GNU Free Documentation License
|
college_math.probability
|
.343,.441, .189, .027.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.2.31
|
Each of the four engines on an airplane functions correctly on a given flight with probability .99 , and the engines function independently of each other. Assume that the plane can make a safe landing if at least two of its engines are functioning correctly. What is the probability that the engines will allow for a safe landing?
|
.999996.
|
GNU Free Documentation License
|
college_math.probability
|
.999996.
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.1.2.13
|
In a horse race, the odds that Romance will win are listed as $2: 3$ and that Downhill will win are $1: 2$. What odds should be given for the event that either Romance or Downhill wins?
|
$11: 4$
|
GNU Free Documentation License
|
college_math.probability
|
11: 4
|
college_math.Grinstead_and_Snells_Introduction_to_Probability
|
exercise.3.1.5
|
There are three different routes connecting city A to city B. How many ways can a round trip be made from A to B and back? How many ways if it is desired to take a different route on the way back?
|
9,6 .
|
GNU Free Documentation License
|
college_math.probability
|
9,6 .
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.1.9
|
Suppose you start with eight pennies and flip one fair coin. If the coin comes up heads, you get to keep all your pennies; if the coin comes up tails, you have to give half of them back. Let $X$ be the total number of pennies you have at the end. Compute $E(X)$.
|
$E(X)=6$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(X)=6
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.3.9
|
Let $Z \sim$ Negative-Binomial $(3,1 / 4)$. Compute $P(Z \leq 2)$.
|
53/512
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
53/512
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.5.7
|
Suppose we repeat a certain experiment 2000 times and obtain a sample average of -5 and a standard error of 17 . In terms of this, specify an interval that is virtually certain to contain the experiment's (unknown) true mean $\mu$.
|
$(-6.1404, -3.8596)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
(-6.1404, -3.8596)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.6.5
|
Suppose $P([0,1])=1$, but $P([1 / n, 1])=0$ for all $n=1,2,3, \ldots$. What must $P(\{0\})$ be?
|
1
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
1
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.6.7
|
Let $X_{1}, X_{2}, \ldots, X_{n+1}$ be i.i.d. with distribution $N(0,1)$. Find a value of $C$ such that
$$
C \frac{X_{1}}{\sqrt{X_{2}^{2}+\cdots+X_{n}^{2}+X_{n+1}^{2}}} \sim t(n) .
$$
|
$C=\sqrt{n}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
C=\sqrt{n}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.3.3
|
Consider flipping two fair coins. Let $X=1$ if the first coin is heads, and $X=0$ if the first coin is tails. Let $Y=1$ if the second coin is heads, and $Y=5$ if the second coin is tails. Let $Z=X Y$. What is the probability function of $Z$ ?
|
$p_{Z}(1)=p_{Z}(5)=1 / 4, p_{Z}(0)=1 / 2$, and $p_{Z}(z)=0$ otherwise
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
p_{Z}(1)=p_{Z}(5)=1 / 4, p_{Z}(0)=1 / 2$, and $p_{Z}(z)=0$ otherwise
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.2.9
|
Suppose $S=\{1,2,3,4\}$, and $P(\{1\})=1 / 12$, and $P(\{1,2\})=1 / 6$, and $P(\{1,2,3\})=1 / 3$. Compute $P(\{1\}), P(\{2\}), P(\{3\})$, and $P(\{4\})$.
|
$P(\{1\})=1 / 12, P(\{2\})=1 / 12, P(\{3\})=1 / 6, P(\{4\})=2 / 3$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(\{1\})=1 / 12, P(\{2\})=1 / 12, P(\{3\})=1 / 6, P(\{4\})=2 / 3
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.1.3
|
Suppose that the lifelengths (in thousands of hours) of light bulbs are distributed $\operatorname{Exponential}(\theta)$, where $\theta>0$ is unknown. If we observe $\bar{x}=5.2$ for a sample of 20 light bulbs, record a representative likelihood function. Why is it that we only need to observe the sample average to obtain a representative likelihood?
|
$L\left(\theta \mid x_{1}, \ldots, x_{20}\right)=\theta^{20} \exp (-(20 \bar{x}) \theta)$ and $\bar{x}$ is a sufficient statistic.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
L\left(\theta \mid x_{1}, \ldots, x_{20}\right)=\theta^{20} \exp (-(20 \bar{x}) \theta)$ and $\bar{x}$ is a sufficient statistic.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.2.9
|
Suppose you simultaneously roll two dice $n$ times and record the outcomes. Based on these values, how would you assess the null hypothesis that the outcome on each die is independent of the outcome on the other?
|
Then there are 36 possible pairs $(i, j)$ for $i, j=1, \ldots, 6$. Let $f_{i j}$ denote the frequency for $(i, j)$ and compute chi-squared statistic, $X^{2}=\sum_{i=1}^{6} \sum_{j=1}^{6}\left(f_{i j}-\right.$ $\left.f_{i} \cdot f_{\cdot j} / n\right)^{2} /\left(f_{i \cdot} \cdot f_{\cdot j} / n\right)$. Compute the P-value $P\left(\chi^{2}(25)>X^{2}\right)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Then there are 36 possible pairs $(i, j)$ for $i, j=1, \ldots, 6$. Let $f_{i j}$ denote the frequency for $(i, j)$ and compute chi-squared statistic, $X^{2}=\sum_{i=1}^{6} \sum_{j=1}^{6}\left(f_{i j}-\right.$ $\left.f_{i} \cdot f_{\cdot j} / n\right)^{2} /\left(f_{i \cdot} \cdot f_{\cdot j} / n\right)$. Compute the P-value $P\left(\chi^{2}(25)>X^{2}\right)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.1.9
|
Suppose a statistical model is given by $\left\{f_{1}, f_{2}\right\}$, where $f_{i}$ is an $N(i, 1)$ distribution. Compute the likelihood ratio $L(1 \mid 0) / L(2 \mid 0)$ and explain how you interpret this number.
|
$L(1 \mid 0) / L(2 \mid 0)=4.4817$, the distribution $f_{1}$ is 4.4817 times more likely than $f_{2}$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
L(1 \mid 0) / L(2 \mid 0)=4.4817$, the distribution $f_{1}$ is 4.4817 times more likely than $f_{2}$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.3.9
|
Suppose you know that the probability distribution of a variable $X$ is either $P_{1}$ or $P_{2}$. If you observe $X=1$ and $P_{1}(X=1)=0.75$ while $P_{2}(X=1)=0.001$, then what would you guess as the true distribution of $X$ ? Give your reasoning for this conclusion.
|
$P_{1}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P_{1}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.6.9
|
Suppose $P([0,1 / 2])=1 / 3$. Must there be some $n$ such that $P([1 / n, 1 / 2])>$ $1 / 4$ ?
|
No
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
No
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.8.2.7
|
Suppose you want to test the null hypothesis $H_{0}: \mu=0$ based on a sample of $n$ from an $N(\mu, 1)$ distribution, where $\mu \in\{0,2\}$. How large does $n$ have to be so that the power at $\mu=2$, of the optimal size 0.05 test, is equal to 0.99 ?
|
$n \geq 4$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
n \geq 4
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.1.5
|
Let $X \sim \operatorname{Geometric}(\theta)$ and $Y \sim \operatorname{Poisson}(\lambda)$. Compute $E(8 X-Y+12)$.
|
$E(8 X-Y+12)=8((1-p) / p)-\lambda+12$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(8 X-Y+12)=8((1-p) / p)-\lambda+12
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.7.5
|
Suppose we are told only that $P(X>x)=1 / x^{2}$ for $x \geq 1$, and $P(X>x)=1$ for $x<1$, but we are not told if $X$ is discrete or continuous or neither. Compute $E(X)$.
|
$E(X)=2$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(X)=2
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.7.1
|
Let $X \sim \operatorname{Bernoulli}(1 / 3)$, and let $Y=4 X-2$. Compute the joint $\operatorname{cdf} F_{X, Y}$.
|
$$
F_{X, Y}(x, y)= \begin{cases}0 & \min [x,(y+2) / 4]<0 \\ 1 / 3 & 0 \leq \min [x,(y+2) / 4]<1 \\ 1 & \min [x,(y+2) / 4] \geq 1\end{cases}
$$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
F_{X, Y}(x, y)= \begin{cases}0 & \min [x,(y+2) / 4]<0 \\ 1 / 3 & 0 \leq \min [x,(y+2) / 4]<1 \\ 1 & \min [x,(y+2) / 4] \geq 1\end{cases}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.3.13
|
Let $X$ and $Y$ be independent, with $X \sim \operatorname{Bernoulli}(1 / 2)$ and $Y \sim \operatorname{Bernoulli}(1 / 3)$. Let $Z=X+Y$ and $W=X-Y$. Compute $\operatorname{Cov}(Z, W)$ and $\operatorname{Corr}(Z, W)$.
|
$\operatorname{Cov}(Z, W)=0, \operatorname{Corr}(Z, W)=0$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\operatorname{Cov}(Z, W)=0, \operatorname{Corr}(Z, W)=0
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.7.4.7
|
Determine Jeffreys' prior for the Bernoulli $(\theta)$ model and determine the posterior distribution of $\theta$ based on this prior.
|
Jeffreys' prior is $\sqrt{n} \theta^{-1 / 2}(1-\theta)^{-1 / 2}$. The posterior distribution of $\theta$ is $\operatorname{Beta}(n \bar{x}$ $+1 / 2, n(1-\bar{x})+1 / 2)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Jeffreys' prior is $\sqrt{n} \theta^{-1 / 2}(1-\theta)^{-1 / 2}$. The posterior distribution of $\theta$ is $\operatorname{Beta}(n \bar{x}$ $+1 / 2, n(1-\bar{x})+1 / 2)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.5.7
|
Suppose that a statistical model is given by the family of $N\left(\mu, \sigma_{0}^{2}\right)$ distributions where $\theta=\mu \in R^{1}$ is unknown, while $\sigma_{0}^{2}$ is known. If our interest is in making inferences about the first quartile of the true distribution, then determine $\psi(\mu)$.
|
$\psi(\mu)=\mu+\sigma_{0} z_{0.25}$, where $z_{0.25}$ satisfies $\Phi\left(z_{0.25}\right)=0.25$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\psi(\mu)=\mu+\sigma_{0} z_{0.25}$, where $z_{0.25}$ satisfies $\Phi\left(z_{0.25}\right)=0.25
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.7.7
|
Suppose $P(W>w)=e^{-5 w}$ for $w \geq 0$ and $P(W>w)=1$ for $w<0$. Compute $E(W)$.
|
$E(W)=1 / 5$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(W)=1 / 5
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.1.3
|
Suppose that an urn contains a proportion $p$ of chips labelled 0 and proportion $1-p$ of chips labelled 1. For a sample of $n=2$, drawn with replacement, determine the distribution of the sample mean.
|
If $Z$ is the sample mean, then $P(Z=0)=p^{2}, P(Z=0.5)=2 p(1-p)$, and $P(Z=1)=(1-p)^{2}$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
If $Z$ is the sample mean, then $P(Z=0)=p^{2}, P(Z=0.5)=2 p(1-p)$, and $P(Z=1)=(1-p)^{2}$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.5.7
|
Suppose a baseball pitcher throws fastballs $80 \%$ of the time and curveballs $20 \%$ of the time. Suppose a batter hits a home run on $8 \%$ of all fastball pitches, and on $5 \%$ of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher's next pitch?
|
0.074
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
0.074
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.4.1
|
Suppose we define a process $\left\{X_{n}\right\}$ as follows. Given $X_{n}$, with probability $3 / 8$ we let $X_{n+1}=X_{n}-4$, while with probability $5 / 8$ we let $X_{n+1}=X_{n}+C$. What value of $C$ will make $\left\{X_{n}\right\}$ be a martingale?
|
$C=12 / 5$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
C=12 / 5
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.3.11
|
Let $Y \sim \operatorname{Binomial}(10, \theta)$. Compute $P(Y=10)$.
|
$\theta^{10}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\theta^{10}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.1.11
|
Suppose we have a statistical model $\left\{f_{\theta}: \theta \in[0,1]\right\}$ and we observe $x_{0}$. Is it true that $\int_{0}^{1} L\left(\theta \mid x_{0}\right) d \theta=1$ ? Explain why or why not.
|
No
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
No
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.6.3
|
Let $X \sim N(3,5)$ and $Y \sim N(-7,2)$ be independent. Find values of $C_{1} \neq$ $0, C_{2}, C_{3} \neq 0, C_{4}, C_{5}$ so that $C_{1}\left(X+C_{2}\right)^{2}+C_{3}\left(Y+C_{4}\right)^{2} \sim \chi^{2}\left(C_{5}\right)$.
|
$C_{1}=1 / 5, C_{2}=-3, C_{3}=1 / 2, C_{4}=7, C_{5}=2$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
C_{1}=1 / 5, C_{2}=-3, C_{3}=1 / 2, C_{4}=7, C_{5}=2
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.6.1
|
Let $Z \sim$ Poisson(3). Use Markov's inequality to get an upper bound on $P(Z \geq$ $7)$.
|
$3 / 7$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
3 / 7
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.2.13
|
Explain why it is not possible that the function $\theta^{3} \exp \left(-(\theta-5.3)^{2}\right)$ for $\theta \in R^{1}$ is a likelihood function.
|
A likelihood function cannot take negative values.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
A likelihood function cannot take negative values.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.6.5
|
Let $\{N(t)\}_{t \geq 0}$ be a Poisson process with intensity $a>0$. Compute (with explanation) the conditional probability $P\left(N_{2.6}=2 \mid N_{2.9}=2\right)$.
|
$P\left(N_{2.6}=2 \mid N_{2.9}=2\right)=(2.6 / 2.9)^{2}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P\left(N_{2.6}=2 \mid N_{2.9}=2\right)=(2.6 / 2.9)^{2}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.5.2.7
|
Suppose that $X \sim \operatorname{Gamma}(3,6)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?
|
The mode is $1 / 3$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The mode is $1 / 3$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.4.5
|
Let $Y=3 X+4$. Compute $m_{Y}(s)$ in terms of $m_{X}$.
|
$m_{Y}(s)=e^{4 s} m_{X}(3 s)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
m_{Y}(s)=e^{4 s} m_{X}(3 s)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.4.7
|
Determine the empirical distribution function based on the sample given below.
\begin{tabular}{|rrrrr|}
\hline 1.06 & -1.28 & 0.40 & 1.36 & -0.35 \\
-1.42 & 0.44 & -0.58 & -0.24 & -1.34 \\
0.00 & -1.02 & -1.35 & 2.05 & 1.06 \\
0.98 & 0.38 & 2.13 & -0.03 & -1.29 \\
\hline
\end{tabular}
Using the empirical cdf, determine the sample median, the first and third quartiles, and the interquartile range. What is your estimate of $F(2)$ ?
|
The sample median is estimated by -0.03 and the estimate of the first quartile is -1.28 , and for the third quartile is 0.98 . Also $\hat{F}(2)=\hat{F}(1.36)=0.90$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The sample median is estimated by -0.03 and the estimate of the first quartile is -1.28 , and for the third quartile is 0.98 . Also $\hat{F}(2)=\hat{F}(1.36)=0.90$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.2.15
|
Suppose basketball teams $A$ and $B$ each have five players and that each member of team A is being "guarded" by a unique member of team B. Suppose it is noticed that each member of team A is taller than the corresponding guard from team B. Does it necessarily follow that the mean height of team $\mathrm{A}$ is larger than the mean height of team B? Why or why not?
|
Yes
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Yes
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.1.9
|
Suppose four fair coins are flipped, and let $Y$ be the number of pairs of coins which land the same way (i.e., the number of pairs that are either both heads or both tails). Compute the exact distribution of $Y$.
|
$p_{Y}(y)=1 / 2$ for $y=1$, 2; otherwise, $p_{Y}(y)=0$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
p_{Y}(y)=1 / 2$ for $y=1$, 2; otherwise, $p_{Y}(y)=0
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.4.11
|
Suppose $X$ has density $f$ and $f(x)>f(y)$ whenever $0<x<1<y<2$. Does it follow that $P(0<X<1)>P(1<X<2)$ ? Explain.
|
Yes
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Yes
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.2.13
|
Suppose darts are randomly thrown at a wall. Let $X$ be the distance (in centimeters) from the left edge of the dart's point to the left end of the wall, and let $Y$ be the distance from the right edge of the dart's point to the left end of the wall. Assume the dart's point is 0.1 centimeters thick, and that $E(X)=214$. Compute $E(Y)$.
|
$E(Y)=214.1$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
E(Y)=214.1
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.3.2.9
|
Suppose $X$ has density function $f(x)=3 / 20\left(x^{2}+x^{3}\right)$ for $0<x<2$, otherwise $f(x)=0$. Compute each of $E(X), E\left(X^{2}\right)$, and $E\left(X^{3}\right)$, and rank them from largest to smallest.
|
Let $\mu_{k}=E\left(X^{k}\right)$, then $\mu_{1}=39 / 25, \mu_{2}=64 / 25, \mu_{3}=152 / 35$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
Let $\mu_{k}=E\left(X^{k}\right)$, then $\mu_{1}=39 / 25, \mu_{2}=64 / 25, \mu_{3}=152 / 35$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.8.1.9
|
Suppose a statistical model comprises all continuous distributions on $R^{1}$. Based on a sample of $n$, determine a UMVU estimator of $P((-1,1))$, where $P$ is the true probability measure. Justify your answer.
|
$n^{-1} \sum_{i=1}^{n} I_{(-1,1)}\left(X_{i}\right)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
n^{-1} \sum_{i=1}^{n} I_{(-1,1)}\left(X_{i}\right)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.3.9
|
A coin was tossed $n=1000$ times, and the proportion of heads observed was 0.51. Do we have evidence to conclude that the coin is unfair?
|
$\mathrm{P}$-value $=0.527$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\mathrm{P}$-value $=0.527
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.4.1
|
Suppose we obtained the following sample from a distribution that we know has its first six moments. Determine an approximate 0.95-confidence interval for $\mu_{3}$.
\begin{tabular}{|rrrrrrrrrr|}
3.27 & -1.24 & 3.97 & 2.25 & 3.47 & -0.09 & 7.45 & 6.20 & 3.74 & 4.12 \\
1.42 & 2.75 & -1.48 & 4.97 & 8.00 & 3.26 & 0.15 & -3.64 & 4.88 & 4.55 \\
\hline
\end{tabular}
|
$m_{3} \pm z_{(1+\gamma) / 2} s_{3} / \sqrt{n}=(26.027,151.373)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
m_{3} \pm z_{(1+\gamma) / 2} s_{3} / \sqrt{n}=(26.027,151.373)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.1.15
|
Suppose we have a quantitative response variable $Y$ and two categorical predictor variables $W$ and $X$, both taking values in $\{0,1\}$. Suppose the conditional distributions of $Y$ are given by
$$
\begin{aligned}
& Y \mid W=0, X=0 \sim N(2,5) \\
& Y \mid W=1, X=0 \sim N(3,5) \\
& Y \mid W=0, X=1 \sim N(4,5) \\
& Y \mid W=1, X=1 \sim N(4,5) .
\end{aligned}
$$
Does $W$ have a relationship with $Y$ ? Does $X$ have a relationship with $Y$ ? Explain your answers.
|
$W$ has a relationship with $Y$ and $X$ has a relationship with $Y$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
W$ has a relationship with $Y$ and $X$ has a relationship with $Y$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.4.7
|
Let $M>0$, and suppose $f(x)=c x^{2}$ for $0<x<M$, otherwise $f(x)=0$. For what value of $c$ (depending on $M$ ) is $f$ a density?
|
$c=3 / M^{3}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
c=3 / M^{3}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.11.4.5
|
Let $\left\{X_{n}\right\}$ be a martingale, with initial value $X_{0}=5$. Suppose we know that $P\left(X_{8}=3\right)+P\left(X_{8}=4\right)+P\left(X_{8}=6\right)=1$, i.e., $X_{8}$ is always either 3 , 4, or 6 . Suppose further that $P\left(X_{8}=3\right)=2 P\left(X_{8}=6\right)$. Compute $P\left(X_{8}=4\right)$.
|
$P\left(X_{n}=4\right)=5 / 8$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P\left(X_{n}=4\right)=5 / 8
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.6.11
|
Let $X$ have density function $f_{X}(x)=(1 / 2) \sin (x)$ for $0<x<\pi$, otherwise $f_{X}(x)=0$. Let $Y=X^{2}$. Compute the density function $f_{Y}(y)$ for $Y$.
|
$f_{Y}(y)=y^{-1 / 2} \sin \left(y^{1 / 2}\right) / 4$ for $0<y<\pi^{2}$ and 0 otherwise
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
f_{Y}(y)=y^{-1 / 2} \sin \left(y^{1 / 2}\right) / 4$ for $0<y<\pi^{2}$ and 0 otherwise
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.6.13
|
Let $X \sim \operatorname{Normal}(0,1)$. Let $Y=X^{3}$. Compute the density function $f_{Y}(y)$ for $Y$.
|
$f_{Y}(y)=(2 \pi)^{-1 / 2}\left(3|y|^{2 / 3}\right)^{-1} \exp \left(-|y|^{2 / 3} / 2\right)$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
f_{Y}(y)=(2 \pi)^{-1 / 2}\left(3|y|^{2 / 3}\right)^{-1} \exp \left(-|y|^{2 / 3} / 2\right)
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.10.3.11
|
Suppose that $X$ and $Y$ are random variables such that a regression model describes the relationship between $Y$ and $X$. If $E(Y \mid X)=\beta_{1}+\beta_{2} X^{2}$, then discuss whether or not this is a simple linear regression model (perhaps involving a predictor other than $X$ ).
|
We can write $E(Y \mid X)=E\left(Y \mid X^{2}\right)$ in this case and $E\left(Y \mid X^{2}\right)=\beta_{1}+\beta_{2} X^{2}$, so this is a simple linear regression model but the predictor is $X^{2}$ not $X$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
We can write $E(Y \mid X)=E\left(Y \mid X^{2}\right)$ in this case and $E\left(Y \mid X^{2}\right)=\beta_{1}+\beta_{2} X^{2}$, so this is a simple linear regression model but the predictor is $X^{2}$ not $X$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.6.9
|
Let $X \sim N(3,5)$ and $Y \sim N(-7,2)$ be independent. Find values of $C_{1}, C_{2}, C_{3}$, $C_{4}, C_{5}, C_{6}, C_{7}$ so that
$$
\frac{C_{1}\left(X+C_{2}\right)^{C_{3}}}{\left(Y+C_{4}\right)^{C_{5}}} \sim F\left(C_{6}, C_{7}\right) .
$$
|
$C_{1}=2 / 5, C_{2}=-3, C_{3}=2, C_{4}=7, C_{5}=2, C_{6}=1, C_{7}=1$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
C_{1}=2 / 5, C_{2}=-3, C_{3}=2, C_{4}=7, C_{5}=2, C_{6}=1, C_{7}=1
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.5.3
|
Describe a Monte Carlo approximation of $\int_{0}^{\infty} e^{-5 x-14 x^{2}} d x$. (Hint: Remember the Exponential(5) distribution.)
|
This integral equals $(1 / 5) E\left(e^{-14 Z^{2}}\right)$, where $Z \sim$ Exponential(5).
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
This integral equals $(1 / 5) E\left(e^{-14 Z^{2}}\right)$, where $Z \sim$ Exponential(5).
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.4.5.1
|
Describe a Monte Carlo approximation of $\int_{-\infty}^{\infty} \cos ^{2}(x) e^{-x^{2} / 2} d x$.
|
The integral equals $\sqrt{2 \pi} E\left(\cos ^{2}(Z)\right)$, where $Z \sim N(0,1)$.
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
The integral equals $\sqrt{2 \pi} E\left(\cos ^{2}(Z)\right)$, where $Z \sim N(0,1)$.
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.1.5.5
|
Suppose we deal five cards from an ordinary 52-card deck. What is the conditional probability that the hand contains all four aces, given that the hand contains at least four aces?
|
1
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
1
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.2.3.19
|
Suppose an urn contains 1000 balls - one of these is black, and the other 999 are white. Suppose that 100 balls are randomly drawn from the urn with replacement. Use the appropriate Poisson distribution to approximate the probability that five black balls are observed.
|
$P(X=5) \approx\left((100 / 1000)^{5} / 5\right.$ ! $) \exp \{-100 / 1000\}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
P(X=5) \approx\left((100 / 1000)^{5} / 5\right.$ ! $) \exp \{-100 / 1000\}
|
college_math.Probability_and_Statistics-The_Science_of_Uncertainty
|
exercise.6.2.5
|
If $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from a $\operatorname{Gamma}\left(\alpha_{0}, \theta\right)$ distribution, where $\alpha_{0}>0$ and $\theta \in(0, \infty)$ is unknown, then determine the MLE of $\theta$.
|
$\hat{\theta}=\alpha_{0} / \bar{x}$
|
The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.
|
college_math.probability
|
\hat{\theta}=\alpha_{0} / \bar{x}
|
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