MA212-2023秋-期中(回忆版)

1 选择题

1-1

Event $A$ and event $B$ are independent, and the probability that at least one of $A,B$ occurs is $8/9$ .It is known that the probability that $A$ occurs and $B$ does not occur is equal to the probability that $A$ does not occur and $B$ occurs, then the probability of $A$ , i.e., $P(A)$ , equals( )

(A) $\frac{2}{3}$

(B) $\frac{1}{3}$

(C) $\frac{4}{9}$

(D) $\frac{5}{9}$

1-2

If $F(x)=\frac{2}{2+x^2}$ is the cumulative distribution function of random variable $X$ , then the value range of $X$ is $()$

(A) $(-\mathrm{\infty },\mathrm{\infty })$

(B) $(0,\mathrm{\infty })$

(C) $(-\mathrm{\infty },0)$

(D) $(0,1)$

1-3

Random variable $X$ follows an exponential distribution with parameter 1, i.e., $X\sim$ $\mathrm{Exp}(1),a$ is a constant greater than zero, then $P(X\le a+1\mid X>a)$ equals( )

(A) $1-e^{-1}$

(B) $e^{-1}$

(C) $1-e^{-a}$

(D) $e^{-a}$

1-4

$D$ is a flat area enclosed by the curve $y=\frac{1}{x}$ and straight lines $y=0,x=1,x=e^2$ . The two-dimensional random variable $(X,Y)$ follows a uniform distribution on $D$ , then the value of the marginal density function of $(X,Y)$ for $X$ at $X=2$ , i.e., $f_X(2)=()$

(A) $\frac{1}{2}$

(B) $\frac{1}{4}$

(C) $\frac{1}{e}$

(D) $\frac{1}{e^2}$

1-5

Random variables $X$ and $Y$ are independent, $X\sim N(0,1),Y\sim N(1,1)$ , then which of the following statements is correct( )

(A) $P(X-Y\le 0)=0.5$

(B) $P(X-Y=1)=0.5$

(C) $P(X+Y\le 0)=0.5$

(D) $P(X+Y\le 1)=0.5$

2 填空题

2-1

Let $A,B,C$ be three mutually independent events, with $P(A)=P(B)=\frac{1}{2},P(C)=\frac{1}{3}$ . Then the probability that at most one of the events $A,B,C$ happens is

2-2

Distribute randomly 6 chopsticks from 3 different pairs to three persons.Then the probability of at least one person getting two chopsticks from the same pair is

2-3

Take two balls without replacement from 3 red balls, 2 black balls and 1 white ball. Then the probability that both balls are red is $\underset{―}{}$

2-4

Let $X$ be a Poisson random, variable with parameter $\lambda =\ln 2$ and let $F(x)$ be its cumulative distribution function.Then $F(1)=$ $\underset{―}{}$

2-5

Toss a fair coin 10 times. Let $A$ be the event of getting exactly 2 heads in total, and let $B$ be the event of getting all tails in the first 5 tosses. Then $P(B\mid A)=$ $\underset{―}{}$

2-6

Suppose that $X$ is a random variable with cumulative distribution function $F(x)=a{e}^{bx}/(1+2e^x)$. Then $a=$ $\underset{―}{}$, $b=$ $\underset{―}{}$

2-7

In an exam that requires 60 to pass, suppose that the score of student $A$ has the normal distribution $N(65,a^2)$ and the score of student $B$ has the normal distribution $N(60+a,{10}^2)$. If the probability of $A$ passing the exam is at least as high as $B$ doing so, then the range for $a$ is $\underset{―}{}$

2-8

Suppose that $X$ has the uniform distribution over $(0,1)$, and given $X$, $Y$ has the uniform distribution over $(0,X)$. Consider the events $A=\{Y\ge \frac{1}{3}\}$ and $B=\{X\le \frac{2}{3}\}$

Then $P(B\mid A)=$ $\underset{―}{}$

2-9

Two independent random variables $X$ and $Y$ satisfy that $X,Y\in \{0,1\}$ and $P(X=0$, $Y=1)=\frac{1}{4},P(X=1,Y=0)=\frac{1}{6}$. Then $P(X=0,Y=0)=$ $\underset{―}{}$

2-10

Let $X$ and $Y$ be independent and be exponential random variables with parameters 1 and 2, respectively. Let $Z=max(X,Y)$. Then for $z\ge 0$, the density function for $Z$ is $f_Z(z)=$ $\underset{―}{}$

3 解答题

3-1

Let the sample space be $\mathrm{\Omega }$. Now consider two events $A$ and $B$, with the probability of occurrence being respectively 0.4 and 0.7

(1) Are $A$ and $B$ disjoint? Prove your answer;

(2) If $A\cup B=\mathrm{\Omega }$, are $A$ and $B$ independent? Prove your answer;

(3) Now consider a geometric model of probability. Let $\mathrm{\Omega }=[0,1]$ be the unit interval in $\mathbb{R}$, and suppose $A=[0,0.4]$, $B=[a,b]$, $0\le a\le b\le 1$. If $A$ and $B$ independent, find the values of $a$ and $b$

3-2

An insurance company divides the insured into three categories: "cautious", "average" and "rash". Statistics show that the probabilities of accidents for these three types of people within a year are 0.05, 0.15 and 0.3 respectively. If "cautious" insured people account for 20%, "average" people account for 50%, and "rash" people 30%

(1) What is the probability that an insured person will be involved in an accident within a year?

(2) If it is known that an insured person has been involved in an accident, find the probability that he belongs to the "cautious" category

3-3

Suppose that the random variable $X$ follows normal distribution, $X\sim N(5,25)$

(1) Compute $P(3<X<7)$;

(2) Find $x$ such that $P(X>x)=0.39$;

(3) Find the value range and the density function of $Y=e^X$

[Note: Standard normal distribution table $\mathrm{\Phi }(0.26)=0.6$; $\mathrm{\Phi }(0.28)=0.61$; $\mathrm{\Phi }(0.30)=0.62$; $\mathrm{\Phi }(0.33)=0.63$; $\mathrm{\Phi }(0.36)=0.64$; $\mathrm{\Phi }(0.39)=0.65$; $\mathrm{\Phi }(0.4)=0.66$; $\mathrm{\Phi }(0.44)=0.67$]

3-4

Let the joint frequency function of the two-dimensional random variable $(X,Y)$ be

X\Y	0	1	2
0	0.06	0.15	α
1	β	0.35	0.21

(1) What conditions do you need to meet for the constant $\alpha$ and $\beta$?

(2) If $X$ and $Y$ are independent, please calculate the values of $\alpha$ and $\beta$;

(3) If $X$ and $Y$ are independent, please find the marginal frequency functions of $X$ and $Y$, respectively

3-5

Let $X$ and $Y$ have the joint density function:

$$f(x,y)=\{\begin{array}{l}1,|x|<y,0<y<1\\ 0,\text{otherwise}\end{array}$$

(1) Find the marginal densities $f_X(x)$ and $f_Y(y)$

(2) Are $X$ and $Y$ independent?

(3) Find the conditional densities function $f_{X\mid Y}(x\mid y)$ and $f_{Y\mid X}(y\mid x)$

3-6

Suppose that the random variables $X$ and $Y$ are independent, where $X\sim U(0,1)$, $Y\sim U(0,2)$

(1) Find the probability of $P(X\le Y)$

(2) Find the distribution function of $X+Y$