Question: What is a quadratic form?

Choices:
- A quadratic form is any polynomial function whose highest order term is 2.
- A quadratic form is a quadratic homogeneous function with $n$ variables $x_{1}, x_{2}, \cdots, x_{n}$, which can be expressed in the form $f=\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A x}$, where $\boldsymbol{A}$ is a symmetric matrix.
- A quadratic form is a system of linear equations, which contains $n$ variables $x_{1}, x_{2}, \cdots, x_{n}$.
- A quadratic form is a system of inequalities, which contains $n$ variables $x_{1}, x_{2}, \cdots, x_{n}$.


Question: What are the characteristics of matrix $\boldsymbol{A}$ in the quadratic form $f=\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A x}$?

Choices:
- Matrix $\boldsymbol{A}$ is a diagonal matrix.
- Matrix $\boldsymbol{A}$ is a skew-symmetric matrix, i.e., $\boldsymbol{A} = -\boldsymbol{A}^{\top}$.
- Matrix $\boldsymbol{A}$ is an upper triangular matrix.
- Matrix $\boldsymbol{A}$ is a symmetric matrix, i.e., $\boldsymbol{A} = \boldsymbol{A}^{\top}$.


Question: How is the rank of a quadratic form defined?

Choices:
- The rank of a quadratic form is defined as the number of eigenvalues of its corresponding symmetric matrix $\boldsymbol{A}$.
- The rank of a quadratic form is defined as the rank of its corresponding symmetric matrix $\boldsymbol{A}$.
- The rank of a quadratic form is defined as the number of non-zero elements in its corresponding symmetric matrix $\boldsymbol{A}$.
- The rank of a quadratic form is defined as the value of the determinant of its corresponding symmetric matrix $\boldsymbol{A}$.


Question: What is the standard form of a quadratic form?

Choices:
- The standard form of a quadratic form refers to the sum of the coefficients of each term after the quadratic form is expanded.
- If there exists an invertible linear transformation that makes the quadratic form contain only square terms, then this form containing only square terms is called the standard form of a quadratic form.
- The standard form of a quadratic form refers to the quadratic form being decomposable into the product of two linear forms.
- The standard form of a quadratic form refers to the form of the quadratic form containing cross terms.


Question: What is the canonical form of a quadratic form?

Choices:
- The canonical form of a quadratic form refers to its representation as a complete square form.
- The canonical form of a quadratic form means that the coefficients of all variables in the quadratic form are $1$.
- If the coefficients of a quadratic form in its standard form can only be $1$, $-1$, or $0$, then this standard form is called the canonical form.
- The canonical form of a quadratic form means that the coefficients of all variables in the quadratic form are equal.


Question: What transformation can be used to reduce the quadratic form $\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A x}$ to its standard form?

Choices:
- Using a linear transformation $\boldsymbol{x}=\boldsymbol{A y}$.
- Using an orthogonal transformation $\boldsymbol{x}=\boldsymbol{P y}$.
- Using an oblique transformation $\boldsymbol{x}=\boldsymbol{B y}$.
- Using a similarity transformation $\boldsymbol{x}=\boldsymbol{A}^{-1} \boldsymbol{y}$.


Question: In the process of turning a real symmetric matrix $\boldsymbol{A}$ into a diagonal matrix through orthogonal transformation, what are the elements of the diagonal matrix?

Choices:
- The value of the trace of the real symmetric matrix $\boldsymbol{A}$.
- The value of the determinant of the real symmetric matrix $\boldsymbol{A}$.
- The eigenvectors of the real symmetric matrix $\boldsymbol{A}$.
- The eigenvalues of the real symmetric matrix $\boldsymbol{A}$.


Question: What method might we need to use when orthogonally diagonalizing a real symmetric matrix $\boldsymbol{A}$ if there are repeated eigenvalues?

Choices:
- Jacobi method.
- Gaussian elimination.
- Schmidt orthogonalization.
- Power method.


Question: What should the column vectors of the orthogonal matrix $\boldsymbol{P}$ consist of?

Choices:
- The column vectors of the original matrix $\boldsymbol{A}$.
- The row vectors of the matrix $\boldsymbol{A}$.
- The column vectors of the identity matrix $\boldsymbol{E}$.
- Vectors after orthonormalization of eigenvectors.


Question: How does the orthogonal matrix $\boldsymbol{P}$ operate with the real symmetric matrix $\boldsymbol{A}$ to obtain a diagonal matrix?

Choices:
- Through $\boldsymbol{P} \boldsymbol{A} - \boldsymbol{P}$.
- Through $\boldsymbol{P} \boldsymbol{A} \boldsymbol{P}}$.
- Through $\boldsymbol{P} + \boldsymbol{A}$.
- Through $\boldsymbol{P}^{-1} \boldsymbol{A P}$.


Question: What is the congruence of matrices?

Choices:
- If there exists an invertible matrix $\boldsymbol{C}$ such that $\boldsymbol{B}=\boldsymbol{C} \boldsymbol{A}$, then matrix $\boldsymbol{A}$ and $\boldsymbol{B}$ are said to be congruent.
- If there exists an invertible matrix $\boldsymbol{C}$ such that $\boldsymbol{B}=\boldsymbol{C}^{\mathrm{T}} \boldsymbol{A C}$, then matrix $\boldsymbol{A}$ and $\boldsymbol{B}$ are said to be congruent.
- If there exists an invertible matrix $\boldsymbol{C}$ such that $\boldsymbol{B}=\boldsymbol{C} \boldsymbol{A C}$, then matrix $\boldsymbol{A}$ and $\boldsymbol{B}$ are said to be congruent.
- If there exists an invertible matrix $\boldsymbol{C}$ such that $\boldsymbol{B}=\boldsymbol{A C}$, then matrix $\boldsymbol{A}$ and $\boldsymbol{B}$ are said to be congruent.


Question: What property of quadratic forms does the inertia theorem tell us about?

Choices:
- The total sum of the coefficients in the standard form of a quadratic form is invariant, not changing with transformation.
- The product of the coefficients in the standard form of a quadratic form is invariant, not changing with transformation.
- The number of zero coefficients in the standard form of a quadratic form is invariant, not changing with transformation.
- The number of positive and negative coefficients (i.e., the positive and negative inertia indices) in the standard form of a quadratic form is invariant, not changing with transformation.


Question: If the quadratic form $f=\boldsymbol{x}^{\top} \boldsymbol{A x}$ changes to $f=k_{1} y_{1}^{2}+\cdots+k_{r} y_{r}^{2}$ and $f=\lambda_{1} z_{1}^{2}+\cdots+\lambda_{r} z_{r}^{2}$, does the number of positive coefficients remain the same in these two forms?

Choices:
- No, because the sign of the coefficients might change.
- No, the number of positive coefficients is different.
- Yes, but only when the transformation is orthogonal.
- Yes, the number of positive coefficients is the same.


Question: What is the number of positive coefficients in the standard form of a quadratic form called?

Choices:
- Called the positive inertia index of the quadratic form.
- Called the rank of the quadratic form.
- Called the trace of the quadratic form.
- Called the determinant of the quadratic form.


Question: What is the term for the number of negative coefficients in the standard form of a quadratic form?

Choices:
- Called the rank of the quadratic form.
- Called the trace of the quadratic form.
- Called the negative inertia index of the quadratic form.
- Called the determinant of the quadratic form.


Question: What is the necessary and sufficient condition for two real symmetric matrices to be congruent?

Choices:
- They have the same rank and the same number of positive and negative inertia indexes.
- They have the same trace and the same determinant.
- They have the same eigenvalues.
- They have the same eigenvectors.


Question: What is a positive definite quadratic form?

Choices:
- If for every non-zero vector $\boldsymbol{x}$, it holds that $f(\boldsymbol{x})=x^{\top} A x>0$, then the quadratic form $f$ is called positive definite, and the corresponding symmetric matrix $\boldsymbol{A}$ is also called positive definite.
- If for every non-zero vector $\boldsymbol{x}$, it holds that $f(\boldsymbol{x})=x^{\top} A x\geq0$, then the quadratic form $f$ is called positive definite.
- If for every non-zero vector $\boldsymbol{x}$, it holds that $f(\boldsymbol{x})=x^{\top} A x=0$, then the quadratic form $f$ is called positive definite.
- If for every non-zero vector $\boldsymbol{x}$, it holds that $f(\boldsymbol{x})=x^{\top} A x<0$, then the quadratic form $f$ is called positive definite.


Question: What is the necessary and sufficient condition for a quadratic form $f=\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}$ to be positive definite?

Choices:
- The standard form has at least one of its $n$ coefficients equal to zero.
- All $n$ coefficients of its standard form are non-positive.
- All $n$ coefficients of its standard form are negative.
- All $n$ coefficients of its standard form are positive, i.e., all $n$ coefficients of its canonical form are 1, that is, its positive inertia index is equal to $n$.


Question: What is the necessary and sufficient condition for a symmetric matrix $A$ to be positive definite?

Choices:
- All eigenvalues of matrix $A$ are positive.
- All eigenvalues of matrix $A$ are non-positive.
- All eigenvalues of matrix $A$ are zero.
- At least one eigenvalue of matrix $A$ is positive.


Question: What is the Hurwitz theorem?

Choices:
- A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that all the leading principal minors of $A$ are negative.
- A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that any leading principal minor of $A$ is positive.
- A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that all the leading principal minors of $A$ are positive.
- A necessary and sufficient condition for a symmetric matrix $A$ to be positive definite is that at least one of the leading principal minors of $A$ is positive.


Question: What is the necessary and sufficient condition for the symmetric matrix $A$ to be negative definite?

Choices:
- All leading principal minors of the symmetric matrix $A$ are positive.
- The odd-order leading principal minors of the symmetric matrix $A$ are negative, while the even-order leading principal minors are positive.
- The even-order leading principal minors of the symmetric matrix $A$ are negative, while the odd-order leading principal minors are positive.
- All leading principal minors of the symmetric matrix $A$ are negative.


Question: $P(\varnothing)=$

Choices:
- 1
- 0
- 0.5
- -1

College--Probability and Statistics--Random Events and Probability


Question: $P(\bar{A})=$

Choices:
- $1-P(A)$
- $1$
- $P(A)-1$
- $P(A)$

College--Probability and Statistics--Random Events and Probability


Question: When $A_{1}, A_{2}, \cdots, A_{n}$ are mutually exclusive events, $P\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right)=$

Choices:
- $\prod_{i=1}^{n} P\left(A_{i}\right)$
- $\sum_{i=1}^{n} P\left(A_{i}\right)$
- $P(A_n)$
- $P(A_1)$

College--Probability and Statistics--Random Events and Probability


Question: $P(A \cup B)=$

Choices:
- $P(A)-P(B)$
- $P(A B)$
- $P(A)+P(B)$
- $P(A)+P(B)-P(A B)$

College--Probability and Statistics--Random Events and Probability


Question: When $P(A)>0$, $P(A B)=$

Choices:
- $ P(A)$
- $P(B \mid A) P(A)$
- $ P(B)$
- $P(B \mid A) $

College--Probability and Statistics--Random Events and Probability


Question: $B_{1}, B_{2}, \cdots, B_{n}$ are a partition of the entire sample space, and $P\left(B_{i}\right)>0, i=1,2, \cdots, n$, $P\left(A \mid B_{1}\right) P\left(B_{1}\right)+P\left(A \mid B_{2}\right) P\left(B_{2}\right)+\cdots$ $+P\left(A \mid B_{n}\right) P\left(B_{n}\right)=$<<<Answer>>>

Choices:
- $P(A)-P(B)$
- $P(A)$
- $P(B)$
- $P(AB)$

College--Probability and Statistics--Random Events and Probability


Question: $P(A)=P\left(A \mid B_{1}\right) P\left(B_{1}\right)+P\left(A \mid B_{2}\right) P\left(B_{2}\right)+\cdots$ $+P\left(A \mid B_{n}\right) P\left(B_{n}\right)$, where $B_{1}, B_{2}, \cdots, B_{n}$ are a partition of the entire sample space, and $P\left(B_{i}\right)>0, i=1,2, \cdots, n$, what is this formula called?

Choices:
- Multiplication Formula
- Addition Formula
- Bayes' Formula
- Law of Total Probability

College--Probability and Statistics--Random Events and Probability


Question: $B_{1}, B_{2}, \cdots, B_{n}$ are partitions of the entire sample space, and $P\left(B_{i}\right)>0, i=1,2, \cdots, n$, $\frac{P\left(A \mid B_{i}\right) P\left(B_{i}\right)}{\sum_{j=1}^{n} P\left(A \mid B_{j}\right) P\left(B_{j}\right)}=$

Choices:
- $P\left(B_{i} \mid A\right)$
- $P(AB)$
- $P(A)$
- $P(B_i)$

College--Probability and Statistics--Random Events and Probability


Question: $P(B_{i} | A)=\frac{P(A | B_{i}) P(B_{i})}{\sum_{j=1}^{n} P(A | B_{j}) P(B_{j})}, i=1,2, \cdots, n$, where $B_{1}, B_{2}, \cdots, B_{n}$ are a partition of the entire sample space, and $P(A)>0, P(B_{i})>0$, $i=1,2, \cdots, n$, what is this formula called?

Choices:
- Formula of Total Probability
- Bayes' Theorem
- Multiplication Rule
- Addition Rule

College--Probability and Statistics--Random Events and Probability


Question: $A \cup B=$

Choices:
- $B$
- $A$
- $B \cap A$
- $B \cup A$


Question: $A \cap B=$

Choices:
- $B \cup A$
- $B$
- $B \cap A$
- $A$


Question: $A \cup(B \cup C)=$

Choices:
- $(A \cap B) \cap C$
- $(A \cup B) \cup C$
- $B \cap A$
- $B \cup A$


Question: $A \cap(B \cap C)=$

Choices:
- $(A \cup B) \cup C$
- $B \cap A$
- $B \cup A$
- $(A \cap B) \cap C$


Question: $A \cup(B \cap C)=$

Choices:
- $(A \cap B) \cap C$
- $(A \cap B) \cup(A \cap C)$
- $(A \cup B) \cup C$
- $(A \cup B) \cap(A \cup C)$


Question: $A \cap(B \cup C)=$

Choices:
- $(A \cup B) \cap(A \cup C)$
- $(A \cap B) \cap C$
- $(A \cup B) \cup C$
- $(A \cap B) \cup(A \cap C)$


Question: $\overline{A \cup B}=$

Choices:
- $(A \cup B) \cap (A \cup C)$
- $\overline A \cup \overline B$
- $\overline A \cap \overline B$
- $(A \cap B) \cup(A \cap C)$


Question: The complement of $A \cap B$

Choices:
- $(A \cap B) \cup(A \cap C)$
- $\overline{A} \cap \overline{B}$
- $(A \cup B) \cap(A \cup C)$
- $\overline{A} \cup \overline{B}$


Question: What are the classical probability model and the geometric probability model?

Choices:
- Geometric probability model
- Unequal probability model
- Equal probability model
- Classical probability model

College--Probability and Statistics--Random Events and Probability


Question: What is the difference between the classical model and the geometric model in that the sample space of the classical model contains only what elements?

Choices:
- One
- A finite number
- An infinite number
- Equally likely

College--Probability and Statistics--Random Events and Probability


Question: The difference between the classical probability model and the geometric probability model lies in, what elements does the sample space of the geometric probability model contain only?

Choices:
- A finite number
- One
- Equally likely
- An infinite number

College--Probability and Statistics--Random Events and Probability


Question: Probability calculation in the classical model: If the total number of elementary events in the sample space is $n$, and event $A$ contains $k_{A}$ of these elementary events, then the probability of event $A$ occurring is $P(A)=$

Choices:
- $\frac{n}{k_{A}}$
- $n$
- $k_{A}$
- $\frac{k_{A}}{n}$

College--Probability and Statistics--Random Events and Probability


Question: Probability calculation in geometric models: Let the sample space be $\Omega, D_{A}$ is a measurable subregion of $\Omega$. Picking a point randomly from $\Omega$, denote the event $A$ as "the point falls within the region $D_{A}$", then the probability of event $A$ happening is $P(A)=$, where $\mu_{A}, \mu_{\Omega}$ are the geometric measures (like length, area, volume, etc.) of $D_{A}$ and $\Omega$ respectively=

Choices:
- $\mu_{\Omega}$
- $\mu_{A}$
- $\frac{\mu_{\Omega}}{\mu_{A}}$
- $\frac{\mu_{A}}{\mu_{\Omega}}$

College--Probability and Statistics--Random Events and Probability


Question: Under the same conditions, independently and repeatedly conducting $n$ random experiments, this type of experiment is called.

Choices:
- Repeated experiment
- Random experiment
- Independent repeated experiment
- Independent experiment

College--Probability and Statistics--Random Events and Probability


Question: Under the same conditions, if an experiment is repeated $n$ times independently and randomly, and the experiment only has two possible outcomes: occurrence or non-occurrence, then we call this series of independent repeated random experiments <<<Answer1>>>

Choices:
- independent repeated experiments
- Bernoulli trial
- $n$ fold Bernoulli trial
- random experiment

College--Probability and Statistics--Random Events and Probability


Question: Given in a trial, the probability of event $A$ occurring is $p(0<p<1)$, then in $n$ repeated Bernoulli trials, the probability of event $A$ occurring exactly $k$ times is

Choices:
- $\mathrm{C}_{n}^{k} p^{k}(k=0,1, \cdots, n)$
- $\mathrm{C}_{n}^{k} (k=0,1, \cdots, n)$
- $\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}(k=0,1, \cdots, n)$
- $\mathrm{C}_{n}^{k} (1-p)^{n-k}(k=0,1, \cdots, n)$

College--Probability and Statistics--Random Events and Probability


Question: Let the sample space of a random experiment be $S. X=X(e)$ is a real-valued single-valued function defined on the sample space $S$. Call $X=X(e)$ as

Choices:
- random variable
- random experiment
- random constant
- sample space

College--Probability and Statistics--Distribution of Random Variables


Question: Let $X$ be a random variable, $x$ be any real number, the function $F(x)=P\{X \leqslant x\},-\infty<x<+\infty$ is called the

Choices:
- distribution function
- probability density function
- random function
- sine function

College--Probability and Statistics--Distribution of Random Variables


Question: A random variable that can take on a finite number of values or countably infinite many values is called

Choices:
- integer random variable
- discrete random variable
- decimal random variable
- continuous random variable

College--Probability and Statistics--Distribution of Random Variables


Question: A random variable that can take either a finite number of values or countably infinite many values is called

Choices:
- the distribution law of continuous random variable $X$
- the distribution law of discrete random variable $Y$
- the distribution law of discrete random variable $X$
- the distribution law of continuous random variable $Y$

College--Probability and Statistics--Distribution of Random Variables


Question: For the distribution function $F(x)$ of a random variable $X$, if there exists a non-negative integrable function $f(x)$, such that for any real number $x$, $F(x)=\int_{-\infty}^{x} f(t) \mathrm{d} t$, then $X$ is referred to as

Choices:
- The distribution law of a discrete random variable $X$
- A continuous random variable
- The distribution law of a continuous random variable $X$
- A discrete random variable

College--Probability and Statistics--Distribution of Random Variables


Question: For the distribution function $F(x)$ of a random variable $X$, if there exists a non-negative integrable function $f(x)$, such that for any real number $x$, $F(x)=\int_{-\infty}^{x} f(t) \mathrm{d} t$, then $f(x)$ is called the <<<Answer1>>> of $X$, also known as the probability density <<<Answer2>>>.

Choices:
- continuous random variable
- discrete random variable
- probability density function
- distribution function

College--Probability and Statistics--Distribution of Random Variables


Question: The distribution function of a continuous random variable is <<<Answer>>> continuous

Choices:
- Not necessarily
- Under certain conditions
- Definitely not
- Inevitably

College--Probability and Statistics--Distribution of Random Variables


Question: The probability density function for a continuous random variable is <<<Answer>>> continuous

Choices:
- under certain conditions
- definitely not
- necessarily
- not necessarily

College--Probability and Statistics--Distribution of Random Variables


Question: The probability mass function of a random variable $X$ is $P\{X=k\}=p^{k}(1-p)^{1-k}, 0<p<1, k=0,1$. If $X$ can only take the values 0 and 1, what distribution does this law of distribution belong to?

Choices:
- Hypergeometric distribution law
- $(0-1)$ distribution law
- Binomial distribution law
- Geometric distribution law

College--Probability and Statistics--Distribution of Random Variables


Question: The $(0-1)$ distribution is also known as

Choices:
- Geometric distribution
- Binomial distribution
- Two-point distribution
- Hypergeometric distribution

College--Probability and Statistics--Distribution of Random Variables


Question: $P\{X=k\}=\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \cdots, n$. What distribution's distribution law is this?

Choices:
- Binomial distribution law
- $(0-1)$ distribution law
- Geometric distribution law
- Hypergeometric distribution law

College--Probability and Statistics--Distribution of Random Variables


Question: In an $n$ repeated Bernoulli trial, the probability of event $A$ occurring is $p$, the random variable $X$ represents the number of times event $A$ occurs, then

Choices:
- $X \sim b(p, n)$
- $X \sim a(n, p)$
- $X \sim b(n)$
- $X \sim b(n, p)$

College--Probability and Statistics--Distribution of Random Variables


Question: What is the relationship between the binomial distribution $b(1, p)$ and the $(0-1)$ distribution when $n=1$?

Choices:
- Different
- Linearly related
- Unrelated
- Identical

College--Probability and Statistics--Distribution of Random Variables


Question: $P\{X=k\}=\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \cdots, n$. What is the distribution law of?

Choices:
- Hypergeometric distribution law
- $(0-1)$ distribution law
- Geometric distribution law
- Binomial distribution law

College--Probability and Statistics--Distribution of Random Variables


Question: $P\{X=k\}=\mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k}, k=0,1$, $2, \cdots, n$. What is the distribution law of?

Choices:
- $(0-2)$ distribution
- Binomial distribution
- Hypergeometric distribution
- Geometric distribution

College--Probability and Statistics--Distribution of Random Variables


Question: What distribution does the law of distribution $P\{X=k\}=\frac{\mathrm{C}_{M}^{k} \mathrm{C}_{N-k}^{n-k}}{\mathrm{C}_{N}^{n}}, k=0,1, \cdots, l$, where $l=\min \{n, M\}, n \leqslant N$ belong to?

Choices:
- Hypergeometric distribution
- $(0-1)$ distribution
- Geometric distribution
- Binomial distribution

College--Probability and Statistics--Distribution of Random Variables


Question: In product sampling inspection (without replacement), assuming there are $M$ defective items among $N$ products, and $n$ products are randomly drawn, if the number of found defective items is denoted by $X$, then what distribution does $X$ follow with parameters $N, M, n$?

Choices:
- Geometric distribution
- Hypergeometric distribution
- $(0-2)$ distribution
- Binomial distribution

College--Probability and Statistics--Distribution of Random Variables


Question: $P\{X=k\}=\frac{\lambda^{k} \mathrm{e}^{-\lambda}}{k !}, k=0,1,2, \cdots$, where $\lambda>0$ is a constant. Which distribution's distribution law is this?

Choices:
- Geometric distribution's distribution law
- Binomial distribution's distribution law
- Poisson distribution's distribution law
- $(0-1)$ distribution's distribution law

College--Probability and Statistics--Distribution of Random Variables


Question:
If the probability density of a continuous random variable $X$ is
$f(x)= \begin{cases}\frac{1}{b-a}, & a<x<b, \\ 0, & \text { otherwise, }\end{cases}$then, $X$ is said to follow what distribution in the interval $(a, b)$?

Choices:
- Exponential distribution
- Gamma distribution
- Uniform distribution
- Normal distribution

College--Probability and Statistics--Distribution of Random Variables


Question:
If the probability density of a continuous random variable $X$ is

$$
f(x)= \begin{cases}\lambda \mathrm{e}^{-\lambda x}, & x>0, \\ 0, & \text { otherwise, }\end{cases}
$$

where $\lambda>0$ is a constant, then $X$ is said to follow a distribution with parameter $\lambda$ of what kind?

Choices:
- Gamma distribution
- Uniform distribution
- Normal distribution
- Exponential distribution

College--Probability and Statistics--Distribution of Random Variables


Question:
A random variable $X$ following an exponential distribution has the following property:
For any $s, t>0$, it holds that
$$
P\{X>s+t \mid X>s\}=P\{X>t\} .
$$
What is this property called?

Choices:
- Memorylessness
- Distributiveness
- Exponentiality
- With memory

College--Probability and Statistics--Distribution of Random Variables


Question:
If the probability density of the continuous random variable $X$ is

$$
f(x)=\frac{1}{\sqrt{2 \pi} \sigma} \mathrm{e}^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}}, \quad-\infty<x<+\infty,
$$

where $\mu, \sigma(\sigma>0)$ are constants, then $X$ is said to follow a distribution with parameters $\mu, \sigma$ of what type?

Choices:
- Uniform distribution
- Exponential distribution
- Normal distribution
- Gamma distribution

College--Probability and Statistics--Distribution of Random Variables


Question: How to find the probability density of $Y=g(X)$ given that the random variable $X$ has a probability density of $f(x)$ through definition?

Choices:
- $F_{Y}(y)=\int_{f(x) \leqslant y} g(x) \mathrm{d} x $
- $F_{Y}(y)=\int_{g(x) \leqslant y} f(x) \mathrm{d} x $
- $g(x)$
- $f(x)$

College--Probability and Statistics--Distribution of Random Variables


Question: Suppose random variable $X$ has probability density $f_{X}(x),-\infty<x<+\infty$, and suppose the function $g(x)$ is differentiable everywhere and always has $g^{\prime}(x)>0$ (or always has $\left.g^{\prime}(x)<0\right)$, then $Y=g(X)$ is a continuous random variable, whose probability density is <<<Answer>>>> where $\alpha=\min \{g(-\infty), g(+\infty)\}, \beta=\max \{g(-\infty), g(+\infty)\}, h(y)$ is the inverse function of $g(x)$.

Choices:
- $f_{X}[h(y)]$
- Answer:$f_{Y}(y)= \begin{cases}f_{X}[h(y)]\left|h^{\prime}(y)\right|, & \alpha<y<\beta, \\ 0, & \text { others, }\end{cases}$
- 0
- Answer:$f_{Y}(y)= \begin{cases}f_{X}[h(y)]\left|h^{\prime}(y)\right|, & \alpha>y>\beta, \\ 0, & \text { others, }\end{cases}$

College--Probability and Statistics--Distribution of Random Variables


Question: In general, suppose $E$ is a random experiment with sample space $S$. If $X=X(e)$ and $Y=Y(e)$ are random variables defined on $S$, the vector $(X, Y)$ formed by them is called what?

Choices:
- One-dimensional random vector
- Three-dimensional random vector
- Two-dimensional random vector
- Four-dimensional random vector

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
Let $(X, Y)$ be a two-dimensional random variable, for any real numbers $x, y$, the bivariate function:

$$
F(x, y)=P\{(X \leqslant x) \cap(Y \leqslant y)\} \xlongequal{\text { denoted by }} P\{X \leqslant x, Y \leqslant y\}
$$

is called <<<Answer1>>>, or known as <<<Answer2>>>.

Choices:
- The sine function of the two-dimensional random variable $(X, Y)$
- The probability density function of the two-dimensional random variable $(X, Y)$
- The random function of the two-dimensional random variable $(X, Y)$
- The distribution function of the two-dimensional random variable $(X, Y)$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: If a two-dimensional random variable $(X, Y)$ can take a finite number of pairs or a countably infinite number of pairs, what is $(X, Y)$ called?

Choices:
- One-dimensional discrete random variable
- One-dimensional continuous random variable
- Two-dimensional continuous random variable
- Two-dimensional discrete random variable

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
Let the two-dimensional discrete random variable $(X, Y)$ take all possible values $(x_{i}, y_{j}), i, j=1,2, \cdots$, and let $P\{X=x_{i}, Y=y_{j}\}=$ $p_{ij}, i, j=1,2, \cdots$. According to the definition of probability,
$$
p_{ij} \geq 0, \quad \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} p_{ij}=1
$$
What do we call $P\{X=x_{i}, Y=y_{j}\}=p_{ij}, i, j=1,2, \cdots$?

Choices:
- The distribution law of the two-dimensional continuous random variable $(X, Y)$
- The distribution law of the two-dimensional discrete random variable $(X, Y)$
- The distribution law of the one-dimensional discrete random variable X
- The distribution law of the one-dimensional discrete random variable Y

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
For the distribution function $F(x, y)$ of a two-dimensional random variable $(X, Y)$, if there exists a non-negative integrable function $f(x, y)$, such that for any $x, y$, it holds that
$$
F(x, y)=\int_{-\infty}^{y} \int_{-\infty}^{x} f(u, v) \mathrm{d} u \mathrm{~d} v,
$$
then what is $(X, Y)$ called?

Choices:
- Two-dimensional continuous random variable
- Two-dimensional discrete random variable
- One-dimensional discrete random variable
- One-dimensional continuous random variable

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
For the distribution function $F(x, y)$ of a two-dimensional random variable $(X, Y)$, if there exists a non-negative integrable function $f(x, y)$, such that for any $x, y$, it holds that
$$
F(x, y)=\int_{-\infty}^{y} \int_{-\infty}^{x} f(u, v) \mathrm{d} u \mathrm{~d} v,
$$
then what is the function $f(x, y)$ called?

Choices:
- The distribution function of the two-dimensional random variable $(X, Y)$
- The distribution function of the one-dimensional random variable $(X, Y)$
- The probability density of the two-dimensional random variable $(X, Y)$
- The probability density of the one-dimensional random variable $(X, Y)$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: The two-dimensional random variable $(X, Y)$, as a whole, has a distribution function $F(x, y)$. Both $X$ and $Y$ are random variables, each with its own distribution function, denoted as $F_{X}(x), F_{Y}(y)$, respectively called the two-dimensional random variable $(X, Y)$'s <<<Answer1>>> and <<<Answer2>>>

Choices:
- marginal distribution function with respect to $Y$, marginal distribution function with respect to $X$
- distribution function with respect to $Y$, distribution function with respect to $X$
- marginal distribution function with respect to $X$, marginal distribution function with respect to $Y$
- distribution function with respect to $X$, distribution function with respect to $Y$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
The marginal distribution function can be determined by the <<<Answer>>> of $(X, Y)$.
$$
F_{X}(x)=P\{X \leqslant x\}=P\{X \leqslant x, Y<+\infty\}=F(x,+\infty) .
$$

Choices:
- probability density function $f(x, y)$
- random function
- sine function
- distribution function $F(x, y)$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
The two-dimensional random variable $(X, Y)$ as a whole has a distribution function $F(x, y)$. Both $X$ and $Y$ are random variables and have their own distribution functions, which are denoted as $F_{X}(x), F_{Y}(y)$, respectively referred to as the two-dimensional random variable $(X, Y)$'s <<<Answer1>>> and <<<
Answer2>>>

Choices:
- distribution law with respect to $Y$, distribution law with respect to $X$
- marginal distribution law with respect to $X$, marginal distribution law with respect to $Y$
- distribution law with respect to $X$, distribution law with respect to $Y$
- marginal distribution law with respect to $Y$, marginal distribution law with respect to $X$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
For the continuous random variables $(X, Y)$, suppose its probability density is $f(x, y)$, define
$$
f_{X}(x)=\int_{-\infty}^{+\infty} f(x, y) \mathrm{d} y, \quad f_{Y}(y)=\int_{-\infty}^{+\infty} f(x, y) \mathrm{d} x,
$$
respectively, $f_{X}(x), f_{Y}(y)$ are called the <<<Answer1>>> and <<<Answer2>>>

Choices:
- probability density with respect to $Y$, probability density with respect to $X$
- marginal probability density with respect to $X$, marginal probability density with respect to $Y$
- marginal probability density with respect to $Y$, marginal probability density with respect to $X$
- probability density with respect to $X$, probability density with respect to $Y$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
Let $(X, Y)$ be a two-dimensional discrete random variable. For a fixed $j$, if $P{Y = y_i} > 0$, then
$$
P\left\{X=x_{i} \mid Y=y_{j}\right\}=\frac{P\left\{X=x_{i}, Y=y_{j}\right\}}{P\left\{Y=y_{j}\right\}}=\frac{p_{i j}}{p_{j}}, i=1,3, \cdots
$$
is called what under the condition of $Y=y_{j}$?

Choices:
- The conditional distribution law of random variable $X$
- The conditional distribution law of random variable $(X, Y)$
- The conditional distribution law of random variable $Y$
- None of the above

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
For a fixed $i$, if $P\left\{X=x_{i}\right\}>0$, then
$$
P\left\{Y=y_{j} \mid X=x_{i}\right\}=\frac{P\left\{X=x_{i}, Y=y_{j}\right\}}{P\left\{X=x_{i}\right\}}=\frac{p_{ij}}{p_{i} .}, j=1,2, \cdots
$$
is referred to as what under the condition $X=x_{i}$?

Choices:
- The conditional distribution law of random variable $Y$
- None of the above
- The conditional distribution law of random variable $(X, Y)$
- The conditional distribution law of random variable $X$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: The probability density of the two-dimensional random variable $(X, Y)$ is $f(x, y)$. The marginal probability density of $(X, Y)$ with respect to $Y$ is $f_{Y}(y)$. If for a fixed $y, f_{Y}(y)>0$, then $\frac{f(x, y)}{f_{Y}(y)}$ is called what under the condition $Y=y$?

Choices:
- Conditional probability
- None of these
- Conditional probability density
- Probability density

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: The probability density of the two-dimensional random variables $(X, Y)$ is $f(x, y)$. The marginal probability density of $(X, Y)$ with respect to $Y$ is $f_{Y}(y)$. If for a fixed $y, f_{Y}(y)>0$, then $\frac{f(x, y)}{f_{Y}(y)}$ is called the conditional probability density of $X$ given $Y=y$, denoted as

Choices:
- $$f_{\mathrm{X|Y}}(x \mid y)=\frac{f(x, y)}{f_{Y}(y)}$$
- $$f_{\mathrm{X|Y}}(x \mid y)=f_{Y}(y$$
- none of the above
- $$f_{\mathrm{X|Y}}(x \mid y)=f(x, y)$$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: What conditions must a conditional probability density satisfy?

Choices:
- None of the above
- $f_{X|Y}(x \mid y)=\frac{f(x, y)}{f_{Y}(y)} \geq 0$
- $f_{X|Y}(x \mid y)=\frac{f(x, y)}{f_{Y}(y)} = 0$
- $f_{X|Y}(x \mid y)=\frac{f(x, y)}{f_{Y}(y)} \leq 0$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
The conditional probability density satisfies the condition: $f_{X|Y}(x \mid y)=\frac{f(x, y)}{f_{Y}(y)} \geqslant 0$;
$$
\int_{-\infty}^{+\infty} f_{X Y}(x \mid y) \mathrm{d} x=\int_{-\infty}^{+\infty} \frac{f(x, y)}{f_{Y}(y)} \mathrm{d} x=\frac{1}{f_{Y}(y)} \int_{-\infty}^{+\infty} f(x, y) \mathrm{d} x=1 .
$$
$$
F_{X|Y}(x \mid y)=P\{X \leqslant x \mid Y=y\}=\int_{-\infty}^{x} \frac{f(x, y)}{f_{Y}(y)} \mathrm{d} x .
$$
What is $\int_{-\infty}^{x} f_{\mathrm{M} Y}(x \mid y) \mathrm{d} x=\int_{-\infty}^{x} \frac{f(x, y)}{f_{Y}(y)} \mathrm{d} x$ under the condition $Y=y$ for $X$ called?

Choices:
- Probability Law
- Conditional Probability Density
- Distribution Function
- Conditional Distribution Function

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: Let $X, Y$ be two independent random variables, with their distribution functions being $F_{X}(x)$ and $F_{Y}(y)$ respectively. What is the distribution function of $M=\max\{X, Y\}$?

Choices:
- None of the above
- $F_{\max }(z) = F_{X}(z) F_{Y}(z)$
- $F_{\max }(z) = F_{X}(z)$
- $F_{\max }(z) = F_{Y}(z)$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: Let $X, Y$ be two independent random variables, with distribution functions $F_{X}(x)$ and $F_{Y}(y)$ respectively. What is the distribution function of $N=\min \{X, Y\}$?

Choices:
- $F_{\min }(z) =P\{N \leqslant z\}=1-[1-F_{Y}(z)]$
- None of the above
- $F_{\min }(z) =P\{N \leqslant z\}=1-[1-F_{X}(z)]$
- $F_{\min }(z) =P\{N \leqslant z\}=1-[1-F_{X}(z)][1-F_{Y}(z)]$

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question: When $X_{1}, X_{2}, \cdots, X_{n}$ are mutually independent and have the same distribution function $F(x)$, what is $F_{\max }(z)$ equal to?

Choices:
- $F(z)^{n}$
- $nF(z)$
- $\frac{F(z)}{n}$
- None of the above

College--Probability and Statistics--Distribution of Multidimensional Random Variables


Question:
Let the probability distribution of a discrete random variable $X$ be

$$
P\left\{X=x_{k}\right\}=p_{k}, \quad k=1,2, \cdots .
$$

If the series $\sum_{k=1}^{\infty} x_{k} p_{k}$ is absolutely convergent, the sum of the series $\sum_{k=1}^{\infty} x_{k} p_{k}$ is called what of the random variable $X$?

Choices:
- Standard deviation
- Variance
- Covariance
- Expected value

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: For a continuous random variable $X$ with probability density $f(x)$, if the integral $\int_{-\infty}^{+\infty} x f(x) \mathrm{d} x$ is absolutely convergent, then the value of the integral $\int_{-\infty}^{+\infty} x f(x) \mathrm{d} x$ is called what of the random variable $X$?

Choices:
- Mathematical Expectation
- Covariance
- Variance
- Standard Deviation

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $X$ is a discrete random variable, its distribution law is $P\left\{X=x_{k}\right\}=p_{k}, k=1,2, \cdots$. If $\sum_{k=1}^{\infty} g\left(x_{k}\right) p_{k}$ is absolutely convergent, then $E(Y)=$

Choices:
- None of the above
- $\sum_{k=1}^{\infty} p_{k}$
- $\sum_{k=1}^{\infty} g\left(x_{k}\right) $
- $\sum_{k=1}^{\infty} g\left(x_{k}\right) p_{k}$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Let $Y$ be a function of the random variable $X$: $Y=g(X)$ (where $g$ is a continuous function). If $X$ is a continuous random variable with probability density $f(x)$, and if $\int_{-\infty}^{+\infty} g(x) f(x) \mathrm{d} x$ is absolutely convergent, then $E(Y)=$

Choices:
- $\int_{-\infty}f(x) \mathrm{d} x$
- $\int_{-\infty} g(x) \mathrm{d} x$
- $f(x)g(x)$
- $\int_{-\infty}^{+\infty} g(x) f(x) \mathrm{d} x$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Assuming that the improper integrals mentioned below are absolutely convergent. If $(X, Y)$ has the joint probability density $f(x, y)$, then $E(X)=$

Choices:
- $\int_{-\infty}^{+\infty} y f_{Y}(y) \mathrm{d} y$
- $\int_{-\infty}^{+\infty} y f_{X}(x) \mathrm{d} y$
- $\int_{-\infty}^{+\infty} x f_{Y}(y) \mathrm{d} x $
- $\int_{-\infty}^{+\infty} x f_{X}(x) \mathrm{d} x $

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Assuming that the improper integrals below are all absolutely convergent. If $(X, Y)$ is the joint probability density of $f(x, y)$, then $E(Y)=$

Choices:
- $\int_{-\infty}^{+\infty} x f_{Y}(y) \mathrm{d} x $
- $\int_{-\infty}^{+\infty} x f_{X}(x) \mathrm{d} x $
- $\int_{-\infty}^{+\infty} y f_{X}(x) \mathrm{d} y$
- $\int_{-\infty}^{+\infty} y f_{Y}(y) \mathrm{d} y$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Assuming that the improper integrals mentioned below are all absolutely convergent. If the joint probability density of $(X, Y)$ is $f(x, y)$ and $Z=g(X, Y)$, then $E(Z)=$

Choices:
- $\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x, y) \mathrm{d} x \mathrm{~d} y$
- $\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x, y) \mathrm{d} x \mathrm{~d} y$
- $\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x, y) f(x, y) \mathrm{d} x \mathrm{~d} y$
- None of these

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: If $C$ is a constant, then $E(C)=$.

Choices:
- $C^2$
- $2C$
- $C$
- $\sqrt{C}$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $X$ is a random variable, $C$ is a constant, then $E(CX)=$.

Choices:
- $CE(X)$
- $C^2E(X)$
- $E(X)$
- $C$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Let $X, Y$ be two random variables, then $E(X+Y)=$.

Choices:
- $E(X)$
- $E(X)+E(Y)$
- $E(Y)$
- $E(X)E(Y)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Given that $X, Y$ are independent random variables, then $E(X Y)=$

Choices:
- $E(X)+E(Y)$
- $E(X)E(Y)$
- $E(X)$
- $E(Y)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Let $X$ be a random variable, if $E\left\{[X-E(X)]^{2}\right\}$ exists, then $E\left\{[X-E(X)]^{2}\right\}$ is called what of $Y$?

Choices:
- Covariance
- Variance
- Mathematical Expectation
- Standard Deviation

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Given $\sigma(X)=\sqrt{D(X)}$, it is called

Choices:
- Covariance
- Standard Deviation
- Variance
- Mathematical Expectation

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: If $C$ is a constant, then $D(C)=$

Choices:
- $C$
- $C^2$
- 0
- $C^2D(x)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: If $X$ is a random variable, $C$ is a constant, then $D(CX)=$

Choices:
- $CD(X)$
- $C^{2}D(X)$
- $C$
- $D(X)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: If $X$ is a random variable, and $C$ is a constant, then $D(X+C)=$

Choices:
- $D(X)$
- $CD(X)$
- $C$
- $C^{2} D(X)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Let $X, Y$ be two random variables, then $D(X+Y)=$

Choices:
- $D(X)+D(Y) $
- $D(X)+D(Y)+2 \operatorname{Cov}(X, Y) $
- $\operatorname{Cov}(X, Y) $
- $D(X)+D(Y)-2 \operatorname{Cov}(X, Y) $

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: The necessary and sufficient condition for $D(X)=0$ is

Choices:
- $P\{X=E(X)\}=1 $
- $P\{X=D(X)\}=1 $
- $P\{X=E(X)\}=0 $
- $P\{X=D(X)\}=0 $

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $E\{[X-E(X)][Y-E(Y)]\}$ is called the _____ of random variables $X$ and $Y$

Choices:
- variance
- mathematical expectation
- standard deviation
- covariance

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $\rho_{X Y}=\frac{\operatorname{Cov}(X, Y)}{\sqrt{D(X)} \sqrt{D(Y)}}$ is called the

Choices:
- mathematical expectation
- variance
- correlation coefficient
- standard deviation

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $\mathrm{Cov}(X, Y)=$

Choices:
- $E(XY)-E(X)E(Y)$
- $D(X)$
- $E(X)E(Y)-E(XY)$
- 0

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $\operatorname{Cov}(X, X)=$

Choices:
- 0
- $D(X)$
- $E(X) E(Y)-E(X Y)$
- $E(X Y)-E(X) E(Y)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $\operatorname{Cov}(X, a)=$

Choices:
- $D(X)$
- $E(X Y)-E(X) E(Y)$
- $E(X) E(Y)-E(X Y)$
- 0

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $D(X \pm Y)=$

Choices:
- $D(X)+D(Y) $
- $D(X)-D(Y) $
- $D(X)+D(Y) \pm 2 \operatorname{Cov}(X, Y)$
- $D(X)-D(Y) \pm 2 \operatorname{Cov}(X, Y)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $\operatorname{Cov}(X, Y)=$

Choices:
- $D(Y, X)$
- $\operatorname{Cov}(Y, X)$
- $D(X)+D(Y) \pm 2 \operatorname{Cov}(X, Y)$
- $E(Y, X)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: When $a, b$ are constants, $\operatorname{Cov}(a X, b Y)=$<<<Answer>>>

Choices:
- $a b \operatorname{Cov}(X, Y)$
- $\operatorname{Cov}(X, Y)$
- $2 a b \operatorname{Cov}(X, Y)$
- $a b$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: $Cov(X_1 \pm X_2, Y) = $

Choices:
- $Cov(X_1, Y) - Cov(X_2, Y)$
- $Cov(X_1, Y)Cov(X_2, Y)$
- $Cov(X_1, Y) \pm Cov(X_2, Y)$
- $Cov(X_1, Y) + Cov(X_2, Y)$

College--Probability and Statistics--Numerical Characteristics of Random Variables


Question: Let the population $X$ have a distribution containing unknown parameters $\theta_{1}, \theta_{2}, \cdots, \theta_{k}$, $X_{1}, X_{2}, \cdots, X_{n}$ are samples from $X$. Suppose the first $k$ moments of the population $X$, $\mu_{l}=E\left(X^{t}\right)(l=1$, $2, \cdots, k)$, exist, using the sample moments $A_{l}=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{l}$ as the estimators for the corresponding population moments $\mu_{t}$, and using continuous functions of sample moments as estimators for continuous functions of population moments, this method of estimation is called

Choices:
- None of the above
- Point estimation
- Maximum likelihood estimation
- Method of moments

College--Probability and Statistics--Mathematical Statistics


Question: The observed value of the method of moments estimator is called

Choices:
- None of the above
- Method of moments estimate
- Point estimate
- Maximum likelihood estimate

College--Probability and Statistics--Mathematical Statistics


Question: Let $X_{1}, X_{2}, \cdots, X_{n}$ be a sample from the population $X$, $x_{1}, x_{2}, \cdots, x_{n}$ be the sample values, $\theta$ be the parameter to be estimated, and $\Theta$ be the possible range of values for $\theta$. The parameter $\hat{\theta}=\hat{\theta}\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ that maximizes the likelihood function $L\left(x_{1}, x_{2}, \cdots, x_{n} ; \theta\right)$ within $\Theta$ is called the maximum likelihood estimate of the parameter $\theta$, and the corresponding statistic $\hat{\theta}=\hat{\theta}\left(X_{1}, X_{2}, \cdots, X_{n}\right)$ is called the maximum likelihood estimator of the parameter $\theta$. This estimation method is called

Choices:
- Maximum likelihood estimation
- Method of moments
- Neither
- Point estimation

College--Probability and Statistics--Mathematical Statistics