instruction stringlengths 12 30k |
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I was wondering if this question should be posted in this forum or in Stack Exchange, but reading both "What kind of questions can I ask here?", in Math I found
> Software that mathematicians use
So I guess it should fit here.
I have a TI-89 (a --powerful, I guess-- Texas Instruments calculator) which I've had... |
Advanced math software for the TI-89 (discriminant of a general polynomial?) |
Let $A$ be a finitely generated Abelian group. Let $tA$ denote the torsion subgroup. Prove that $A$ has a subgroup isomorphic to $A/tA$.
I know that $A/tA$ is torsion free, so my thinking so far has been to take the non-torsion elements of $A$ and consider the subgroup generated by them. Then show that subgroup and... |
Let $g(t)$ be a real function of a real variable such that
$$\sup |p(t)g(t)|\lt \infty$$ for any polynomial $p$. Is it true that for any nonngetive
integer $k$, one has $$\sup_{(x,y)\in\mathbb{R}^2}(\frac{|x|}{1+|y|})^k |g(x-y)|\lt \infty$$
for all real numbers $x,y$ ?
|
If I understand correctly, the Euclidean Dirac operator is given by
$$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$
where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional Clifford algebra with negative-definite signature over the reals (so $e_i^2=-1$), and $x_i$ are the corresponding co... |
How do you show that the Laplacian is the square of the (Euclidean) Dirac operator? |
What are some families of graphs with known <a href="http://en.wikipedia.org/wiki/Tree_decomposition">tree width</a> k?
For instance, for tree k=1, for cycle k=2, for series-parallel graph which is not a tree k=2, what are some other examples?
I'm looking for some easy to construct families (or famous graphs which ... |
All,
Given any number, how can we retrieve each digit of that number?
i.e. For number = 4321, how can I get individual digits 4,3,2 and 1?
|
I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows:
Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its Steifel-Whitney class is $w(\nu)=1$ or $w(\nu)=1+x$. In every case, we need $w(\nu)\cdot w(\tau) = w(\nu \oplus \tau... |
Does having a codimension-1 embedding $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable? |
Let $g(t)$ be a real function of a real variable such that
$$\sup |p(t)g(t)|\lt \infty$$ for any polynomial $p$. Is it true that for any nonngetive
integer $k$, one has $$\sup_{(x,y)\in\mathbb{R}^2}(\frac{|x|}{1+|y|})^k |g(x-y)|\le A_k, $$
for some positive constant $A_k\lt \infty$?
|
This holds for *any* irreducible finite state Markov chain. I'm not sure about a pure combinatorial proof, but it can be shown quite quickly.
Let $M(t)$ be the expected time the chain takes to return to its initial state, if started at $t$. As it spends, on average, a time $M(t)-1$ outside the state $t$ after each v... |
I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".
I don't understand why this might be the case.
In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra. Then we consider the induced fu... |
Is restriction of scalars a pullback? |
I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like.
There are lots of resources for predicate and first order ... |
Resources for learning formal math? |
I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".
I don't understand why this might be the case.
In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra. Then we consider the induced fu... |
Suppose I have a set $\mathcal{S}$ of $N$ distinct items. Now consider the set $\mathcal{P}$ of all possible pairs that I can draw from $S$. Naturally, $|\mathcal{P}| = \binom{N}{2}$. Now when I draw $k$ items (pairs) from $\mathcal{P}$ with a uniform distribution, what is the expected number of distinct items from $S$... |
Since the last value for $j$ is $k^2-1$, none of the terms of the sum are $k$; they are all between $1$ and $k-1$.
How many $1$'s will be in the sum? Well, we'll get $1$ when $j$ is any number between $1^2$ and $2^2-1$; then we'll get $2$ for each number between $2^2$ and $3^2-1$. Then we'll get $3$ for each number ... |
So I found this
$\prod_{n=3}^{inf} (1 - tan( \pi/2^k)^4$
at http://people.missouristate.edu/lesreid/Advanced.html
I have only every done tests for convergence of infinite sums.
At http://en.wikipedia.org/wiki/Infinite_product it shows a way to convert but
in this case an is less than one for all k. I can se... |
So I found this
$$\prod_{n=3}^{\infty} (1 - \tan( \pi/2^k)^4$$
<a href="http://people.missouristate.edu/lesreid/Advanced.html">here</a>.
I have only every done tests for convergence of infinite sums.
At <a href="http://en.wikipedia.org/wiki/Infinite_product">this link</a> it shows a way to convert but
in th... |
So I found this
$$\prod_{k=3}^{\infty} (1 - \tan( \pi/2^k)^4$$
<a href="http://people.missouristate.edu/lesreid/Advanced.html">here</a>.
I have only ever done tests for convergence of infinite sums.
At <a href="http://en.wikipedia.org/wiki/Infinite_product">this link</a> it shows a way to convert but
in thi... |
I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".
I don't understand why this might be the case.
In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra. Then we consider the induced fu... |
Suppose $64$ students are taking an exam. The exam has $16$ questions in multiple choice format. Students must fill an oval for every one of the $16$ questions. They can't revise their answers. The time taken for a students to answer a question is exponentially distributed with mean $8$ minutes. So $f(t) = \frac{1}{8}e... |
Suppose $64$ students are taking an exam. The exam has $16$ questions in multiple choice format. Students must fill an oval for every one of the $16$ questions. They can't revise their answers. The time taken for a students to answer a question is exponentially distributed with mean $8$ minutes. So $f(t) = \frac{1}{8}e... |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
Suppose $f(x,y)$ is bivariate normal. You are given $E(X)$, $E(Y)$, $\text{Var}(X)$ and $\text{Var}(Y)$. You are also given $\rho$ the correlation coefficient. How would one find $\text{Var}(Y|X=k)$? Would you use [the total law of variance][1]? We know that $\rho = \frac{\text{Cov}(X,Y)}{\sigma_x \sigma_y}$.
... |
How to prove that the power **b ^ n** is an irrational number, where **b** is an integer greater than 1 and **n** is an irrational number?
|
Suppose I have a 2-variable function $g(k,n)$ and I know that $g(k,n)=O(n^{f(k)})$, for fixed $k$ as $n \rightarrow \infty$, for some function $f=f(k)$. Suppose I also know that $f(k)=O(\log k)$.
> Is it legitimate to write $g(k,n)=O(n^{O(\log k)})$? |
Is it legitimate to write nested big-Os in an asymptotic formula of multivariable functions? |
I'm dealing with tables that show connectivity within a network. Can anyone point me toward a technique for ordering the rows and columns so that nodes that are connected tend to be close to each other in the table?
For example, if I had a network like this (connections are directional), the structure isn't immediat... |
Technique for "untangling" directed networks? |
Is it legitimate to write nested big-Os in an asymptotic formula for a multivariable function? |
I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started with. I picked up [Schaum's Outline of Discrete Mathematics][1]. I find that some things are alittle hard to understand st... |
Is there a basis-independent proof of Abel's identity? |
[The companion matrix](http://en.wikipedia.org/wiki/Companion_matrix) of a monic polynomial in 1 variable over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules.
It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are b... |
Does having a codimension-1 embedding of a closed manifold $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable? |
I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows:
Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its Steifel-Whitney class is $w(\nu)=1$ or $w(\nu)=1+x$. In every case, we need $w(\nu)\cdot w(\tau) = w(\nu \oplus \tau... |
Say I have an equation where the value is $m-4$ if $m \ge 5, n=3,4$, $n-4$ if $n \ge 5, m=3,4$, and $m+n-9$ if $m,n \ge 5$. How can I combine these all into one statement? How can I combine them with further ones, like $m-6$ where $m \ge 7, n \in [3,6]$, etc? |
Say I have an equation where the value is $m-4$ when $m \ge 5, n=3,4$; $n-4$ when $n \ge 5, m=3,4$; and $m+n-9$ when $m,n \ge 5$. How can I combine these all into one statement? How can I combine them with further ones, like $m-6$ where $m \ge 7, n \in [3,6]$, etc? |
I understand that we use the chain rule to differentiate the $y$ part of it. Like if it stood $xy$, I would use the product rule, that say that the derivative of $y$ is $dy/dx$. But what if it's just $y$ alone or with a constant in front off it?
Thank you! |
$$max{$f(n), $g(n)} = $O($f(n), $g(n))
How do I prove this? Also I'd appreciate the markup being corrected, thanks. |
$\max(f(n), g(n)) = O(f(n), g(n))$
How do I prove this? Also I'd appreciate the markup being corrected, thanks. |
$\max(f(n), g(n)) = O(f(n) + g(n))$
How do I prove this? Also I'd appreciate the markup being corrected, thanks. |
Have you tried [this website][1]? They have a small collection of graphs and a library for computing their exact tree-width, which you can compare against your code.
You can also try constructing "random" graphs by following the definition, though in that case you only know an upper bound on the tree-width; probably... |
I'm very n00b in maths, and I don't understand what I can do with a generating function.
For example, if I want to compute the first spread polynomial, how can I do that from the generating function:
![alt text][1]
or the exponential one:
![alt text][2]
?
Thanks in advance,
rubik
[1]: https://... |
Which are the parallel lines?
![alt text][1]
[1]: https://i.stack.imgur.com/IalyM.jpg
I prove that a and b are parallel but can't prove that c and d are.
The angle ,which is 135 degrees and the 45 one are alternate angles.They aren't equal ,so c *isn't* parallel to d but the answer is that c is parallel to ... |
Let $\lfloor \sqrt{n} \rfloor = a$, then the following sum:
$$\sum_{0\le k < n} \lfloor \sqrt{k} \rfloor = na - \frac{a^3}{3} -
\frac{a^2}{2} - \frac{a}{6},$$
while the slightly more tricky sum:
$$\sum_{0\le k < n} \lfloor k^{1/3} \rfloor = nb - \frac{b^2}{4} - \frac{b^3}{2} - \frac{b^4}{4},$$
where t... |
How to solve this manually ? |
It's best explained with an image of course:
[here (not enough reputation to post an image)][1]
l1, l2, h1 & h2 are all given, no I would like to calclate the length of L3. Is it possible and how?
[1]: http://www.iscriptdesign.com/img/trapezium.gif |
The [group][1] axioms demand that one has an inverse element. When it comes to multiplication zero has no inverse element because one cannot reconstruct the original number after it was multiplied by zero:
There is an irreversible loss of information!
Therefore all numbers cannot form a group under multiplicatio... |
It's best explained with an image of course:
![alt text][1]
L1, L2, h1 & h2 are all given, and I would like to calculate the length of L3. Is it possible, and how?
[1]: https://i.stack.imgur.com/j2MpR.gif
|
How to solve this manually ?
**EDIT:** In my module the answer is given as **$0$** but when I used mathematica `N[Sin[10] + Sin[20] + Sin[30] - Sin[360]]` gives me **$-1.57802$**.So I guess there is some mistake in the problem statement. |
Which are the parallel lines?
![alt text][1]
I prove that a and b are parallel but can't prove that c and d are.
The angle ,which is 135 degrees and the 45 one are alternate angles.They aren't equal ,so c *isn't* parallel to d but the answer is that c is parallel to d.Am I right that they aren't?
**Edit**:
T... |
How to solve this manually ?
**EDIT:** In my module the answer is given as **$0$** but when I used mathematica `N[Sin[10 Degree] + Sin[20 Degree] + Sin[30 Degree] - Sin[360 Degree]]` gives **$1.0156683209925990818958162414 \cdots$**
So I guess there is some mistake in the problem statement. |
How to solve this manually ?
**EDIT:** In my module the answer is given as **$0$** but when I used mathematica `N[Sin[10 Degree] + Sin[20 Degree] + Sin[30 Degree] - Sin[360 Degree],50]` gives **$1.0156683209925990818958162414 \cdots$** (truncated)
So I guess there is some mistake in the problem statement. |
Just as additional information, the inverse of these binomial coefficients satisfy several wonderful identities. For example, on [Mathworld][1] I saw the following amazing identity:
$$\sum_{n \ge 1}\frac{1}{n^k \binom{2n}{n}} = \frac{1}{2}{}_{k+1}F_k(\underbrace{1,\ldots,1}_{k+1}; \frac{3}{2}, \underbrace{2,\ldots,2... |
A power type **b^[(a^/(1/n)]** where **b**, **a** and **n** are positive integers greater than 1 and a^(1/n) is irrational, always represents an irrational number?
|
b^[a^(1/n)]: is it an irrational number? |
How the following conversion is justified ?
$$\cos \biggl( \theta + \frac{2\pi}{3} \biggr) + \cos \biggl( \theta + \frac{4\pi}{3} \biggr) = \cos \biggl( \frac{\pi}{2} - \theta \biggr) - \cos \biggl( \frac{\pi}{3} +\theta \biggr) $$ |
Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line?
What is a parametric equation of such curve given the parametric equation of a curve along which the master goes?
|
Let $A$ be a finite dimensional vector space, $\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $ be an inverse system of finite dimensional vector spaces, and let $f_i:A\to A_i$ be linear maps inducing a map $f:A\to\lim_{\leftarrow} A_i$. If $f$ is an injection (resp. s... |
Let $f:[0,1] \to {\bf R}$ have a following property: $f$ is continuous except at the points $x_{k,n}$, $k=1,\ldots,2^{n-1}$, $n=1,2,3,\ldots$, where $\lim _{x \downarrow x_{k,n} } f(x) = f(x_{k,n} )$ but $f(x_{k,n} ) -
\lim _{x \uparrow x_{k,n} } f(x) = a_n > 0$. Can such a function exist if $a_n > 1/ \log(n)$ for al... |
How much a well-behaved function can jump? |
Let $A$ be a finite dimensional vector space, $\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $ be an inverse system of finite dimensional vector spaces, and let $\phi_i:A\to A_i$ be linear maps inducing a map $\phi:A\to\lim_{\leftarrow} A_i$. If $\phi$ is an injection... |
Let $f:[0,1] \to {\bf R}$ have a following property: $f$ is continuous except at the points $x_{k,n} = \frac{{2k - 1}}{{2^n }}$, $k=1,\ldots,2^{n-1}$, $n=1,2,3,\ldots$, where $\lim _{x \downarrow x_{k,n} } f(x) = f(x_{k,n} )$ but $f(x_{k,n} ) -
\lim _{x \uparrow x_{k,n} } f(x) = a_n > 0$. Can such a function exist if... |
Generalized tractrices (tractories) exist, see [this][1] or [this][2] or [this][3]. (the last two are in French, but with slightly more detail than the first one.)
This [old book][4] ought to be of interest as well.
[1]: http://www.2dcurves.com/derived/generalized%20tractrix.html
[2]: http://www-cabri.imag.f... |
Assume a capacitated network flow problem with graph $G = (N,A)$ and capacities $u_{ij}$ (finite), costs for each path of $c_{ij}$ and I/O of $b_{i}$.
How can it be shown there is an equivalent transportation problem with a source node for every arc $(i,j)$ in $N$ having supply $u_{ij}$, and a sink node with demand ... |
What you're looking for is described on pages 247-249 of Robert Vanderbei's [Linear Programming: Foundations and Extensions][1] (2nd edition), although Vanderbei has some of his sign conventions opposite of yours. Take a close look in particular at Figure 14.4 on page 249 ("Adding a new node to accommodate an arc $(i,... |
I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it's limit.
$\sum\limits_{n=1}^{\infty}\frac{n-sqrt{n^2+n}}{n}$
I've ruled down possible tests to the limit ... |
I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it's limit.
$\sum\limits_{n=1}^{\infty}\frac{sin(n-\sqrt{n^2+n})}{n}$
I've ruled down possible tests to the li... |
I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find it's limit.
$$\displaystyle \sum\limits_{n=1}^{\infty}\dfrac{\sin(n-\sqrt{n^2+n})}{n}$$
I've ruled down possib... |
Won't let me post a picture, so try [http://dev.dosboy.ca/points.png][1].
Sorry if this is a little basic, but my math is pretty rusty. I know the point coordinates of the two blue dots. I also know the distances from the middle blue dot to the edges of the square. How can I calculate the coordinates of the red d... |
![alt text][1]
Sorry if this is a little basic, but my math is pretty rusty. I know the point coordinates of the two blue dots. I also know the distances from the middle blue dot to the edges of the square. How can I calculate the coordinates of the red dot?
[1]: https://i.stack.imgur.com/1jymi.png |
Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... |
Let's use the notation $\{x\} = x - \lfloor x \rfloor$. We can think of $\{\cdot\}$ as a function $\mathbb{R} \longrightarrow \mathbb{R}/\mathbb{Z}$, indeed a homomorphism of groups (with addition as the group operator). That means that $\{x + y\} = \{x\} + \{y\}$, or what matters more in your case, $\{x - y\} = \{x\} ... |
As a physics student, I've come across mathematical objects called **tensors** in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, which I believe are slightly different.
I currently think of them in the following ways, but have a tough time r... |
How much a well-behaved (specifically, càdlàg) function can jump? |
$10 \%$ of people in a population suffer from high blood pressure. Calculate the probability that the twelfth person tested will be the second one with high blood pressure. So this is a negative binomial distribution. Now $P(X=k) = \binom{k+r-1}{k} (1-p)^{r}p^{k}$. So this is the probability of $k$ successes after $r$ ... |
Say we have a set of 12 numbers - (0,1,2,0,3,0,4,0,5,6,0).
We go thru this set one number at a time, and choose a number with a probability 1/3; we sample. What is the probability we end up with 3 0's out of the 4 numbers we will likely choose(on average)?
Is there a closed-form solution for this? |
I have changed the title (replaced "well-behaved" by "càdlàg"), since it seems that "a well-behaved function" might be interpreted as "a function of bounded variation" (rather than "a càdlàg function", which I actually meant).
Let $f:[0,1] \to {\bf R}$ have a following property: $f$ is continuous except at the point... |
How much a càdlàg function can jump? |
so studying for my midterm on Tuesday (intro to abstract algebra). The topics on the exam are Division Algorithm, Divisibility, Prime Numbers, FTA, Congruency, Congruent Classes and very brief introduction to rings.
I was reading a few theorems about Congruency and have a couple of questions.
I want to know wha... |
See the equivalent question on mathoverflow: [What manifolds are bounded by RP^odd?][1]
(Since it seems there is a good reason to have the answer recorded as such (and, borrowing from the suggestion [here][2]), I'm moving my comment here. Since all I'm doing it linking to another place, I don't want to gain reputat... |
Suppose Jill and Jane are individually painting separate rooms. Each of the rooms has $4$ walls. The time taken to paint each wall is exponentially distributed with mean $1$ hour. What is the probability that Jill will finish painting $3$ walls before Jane finishes $2$ walls?
This seems to look somewhat similar to ... |
As a result of <a href="http://math.stackexchange.com/questions/9532/link-between-a-dense-subset-and-a-continuous-mapping">this question</a>, I've been thinking about the following condition on a topological space $Y$:
> For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ ... |
Let $A$ be a finite dimensional vector space, $\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $ be an inverse system of finite dimensional vector spaces, and let $\phi_i:A\to A_i$ be linear maps inducing a map $\phi:A\to\lim\limits_{\leftarrow} A_i$. If $\phi$ is an in... |
Suppose we have an infinite sequence of intervals $[a_n,b_n]$, where for all n, $a_n<b_n$. Then is there some $x\in \mathbb{R}$ such that for each n there is some $k\in\mathbb{Z}$ such that $kx\in [a_n,b_n]$?
It seems unlikely, but I'm not sure where to begin with the proof. Thank you. |
For a sequence of intervals can we find some x such that each interval contains a multiple of x? |
so studying for my midterm on Tuesday (intro to abstract algebra). The topics on the exam are Division Algorithm, Divisibility, Prime Numbers, FTA, Congruency, Congruent Classes and very brief introduction to rings.
I was reading a few theorems about Congruency and have a couple of questions.
I want to know wha... |
Just as additional information, the inverse of these binomial coefficients satisfy several wonderful identities. For example, on [Mathworld][1] I saw the following amazing identity (valid for $k\ge 1$):
$$\sum_{n \ge 1}\frac{1}{n^k \binom{2n}{n}} = \frac{1}{2}{}_{k+1}F_k(\underbrace{1,\ldots,1}_{k+1}; \frac{3}{2}, \... |
Okay, let's start from the beginning. You define the congruence class of $a$ as:
\begin{equation*}
[a]_n = \{ b\in\mathbb{Z}\mid a\equiv a\pmod{n}\}.
\end{equation*}
Now, what does $a\equiv b\pmod{n}$ mean? It means that $n$ divides $a-b$, or equivalently, that $a$ and $b$ leave the same remainder when you divide t... |
Okay, let's start from the beginning. You define the congruence class of $a$ as:
\begin{equation*}
[a]_n = \\{ b\in\mathbb{Z}\mid a\equiv a\pmod{n}\\}.
\end{equation*}
Now, what does $a\equiv b\pmod{n}$ mean? It means that $n$ divides $a-b$, or equivalently, that $a$ and $b$ leave the same remainder when you divide... |
![alt text][1]
[1]: https://i.stack.imgur.com/ULjJd.png
The answers are already given after the = sign of each question. But I dont know how they arrived at these answers. What does it mean to say f(A)=1 and so on? I cant find the connection.. |
How much a càdlàg (i.e., right-continuous with left limits) function can jump? |
The second graph is bipartite: you divide the vertices into two groups and all edges are between the two groups, with no edges between vertices in the same group. Now look at the first graph. And each vertex is exactly adjacent to three vertices on the "other" side.
In the first graph, every vertex is adjacent to ex... |
Suppose $P(A) = 0.6$ and $P(B) = 0.7$. Find the smallest possible value of $P(A \cap B)$. Instead of deducing this from $P(A \cup B) = P(A)+P(B)-P(A \cap B)$, could we just use the fact that $P(A|B)P(B) = P(A \cap B)$ and $P(B|A)P(A) = P(A \cap B)$? I notice that $(0.6)(0.7)^2 = 0.294$ which is close to $0.3$, the smal... |
Let X and Y be independent random variables,
$ P(X = k) = P(Y = k) = p(1 - p)^{k-1} $
How do you show that the pmf of $ Z = X + Y $, is negative binomial, and how do you find
$ P(X = Y) $? |
Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, ... |
If $f\colon[0,1]\to\mathbb{R}$ is cadlag ([continu à droite, limites à gauche][1]) then we can ask what the possible jumps are. That is, for what functions $g\colon(0,1]\mapsto\mathbb{R}$ is there a cadlag function $f$ with $g(x)=\Delta f(x)\equiv f(x)-f(x-)$?
The answer is that $g$ occurs as the jumps... |
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