概率论与数理统计英文题目

适合于学习双语的概率论与数理统计。

Test

1. Consider the set 1,2,3,4,5,6,7,8,9 with subsets

A 1,3,5,7,9 ,

Find the following sets:

(a) A D and B 2,4,6,8 ,C 1,2,3,4 ,D 7,8 A B (b) (C D)c and A (B D)

(E F)c Ec Fc,(b)(E F)c Ec Fc 2. ShowDe Morgans Law: (a)

3.(a)Let P(A)=0.5,P(B)=0.4,P( A∩B)=0.2. Find P(A∪B) and P(A|B)

(b)Consider two fair dice A and B. Die A is six-sided and is numbered 1 through to 6

whilst die B is four-sided and is numbered 1 through 4. Both dice are rolled. Find

the probability of two dice show the same score.

4.(a).Let X be a random variable. X~N( , 2). Show that Y X ~N(0,1)

(b)The pmf of a random variable X which has a Poisson distribution with parameter

2. Find P(X=3).

5. Suppose (X,Y) be bivariate normal distribution.

(X,Y)~N( 1, 12, 2, 22; ). Find the correlation coefficient of X and Y.

6. (a)Let X Uniformly distributed on (-1,1), i.e. X~U(-1,1). Find the Expectation and

Variance of X.

(b)Let X and Y be continuous random variables with joint pdf

fX,Y(x,y)(x,y) R2. Let Z=X+Y, FindfX Y(z). In particular, if X and Y

are independent, Find fX*fY.. 7.(a)The random variable X has pmf is: X 0 1 2 3

PX Find P(1≤X<3) and FX(1.7).

(b)Let X be a continuous random variable with pdf

2e 2xx 0f(x) otherwise 0

Find P(X=3), P(X>1) andP(X 2X 1).

8. X is continuous r.v. with pdf

x 12e f(x) 2

x 1e 2 2x 0 x 0

Let Y X, Find E(Y)

适合于学习双语的概率论与数理统计。

9.Let X be a continuous random variable with probability density function

xe xx 0f(x) otherwise 0

(a) Find P(1<X<2) (b) Find: (1) E(X); ,(2) Given that E(X2)=6. Find Var(X)

10．If the joint probability density function of two random variables is given by

6e 2x 3yx 0,y 0 f(x) otherwise 0

(a) Find the marginal pdf of X and Y.

(b)Determine whether the two random variables are dependent or independent.

11．Suppose the random variable X has the density function

32x 0 f(x) (x 4)3

0otherwise

Find the probability density function of the random variable Y=X+4.

12. Take out a number from 1-200 ( 200 natural numbers).

Find (a) the probability that this number can be drived by 6;

(b) the probability that this number can be drived by 6 as well as 8.

13. Three children are selected at random from a group of five boys and three girls.

(a)What is the probability that all three are boys?

(b)What is the probability at least two girls are selected?

14. (a)Let P(AB) P(AB),P(A)＝p,findP(B). (b)LetP(A) 123，P(A|B) ，P(B|A) .FindP(B). 535

15.(a) Suppose A, B are two events, and P(A)=1/4, P(B)=1/2, P(AB)=1/9, then

evaluate PAB

(b)Use Bayes’ Theorem to show that if P(A),P(B)>0 and p(A) p(A|B) then

P(B) P(BA)

16.(a)Let X has a Binomial distribution with parameters n and p, i.e. X ~ b (n, p).Find

P(X 2)

(b) Show that Cov(X,Y) E(XY) E(X)E(Y)

适合于学习双语的概率论与数理统计。

17. Suppose the distribution function of a random variable X is

0,x 0 F(x) x2,0 x 1，

1,x 1

Find (a) the probability that X gets value within (0.3，0.7);

(b) the density function of X

18. The operational lifetime X, in years, of a battery powered watch has probability density function

cx(6 x)3 x 6f(x) otherwise 0

(a) Find the value of c.

(b) Find the cumulative distribution function of X.

(c) Find the probability that the watch has an operational lifetime in excess of 4 years.

Find a and FX(3.2)

(b) A continuous random variable X having the probability density function

x2

f(x) 3 0

1 x 2elsewhere.Find f(x)dx and P(0<X<1).

20.(a)Let P(BA) P(BA),P(A) 1 P(B).FindP(AB) 3

(b) Suppose X and Y are independent random variables X~B(2,p),Y~B(3,p),and

5P(Y 1) .FindP(Y 1). 9

21. Suppose the density function of (X,Y) is

k 6 x y ,0 x 2,2 y 4f x,y ,else 0

(a) Determine the constant k. (b) Find the probability P{X<1,Y<3}.

22. Let X denote the number of times a certain numerical control machine will malfunction: 1,2,or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is

适合于学习双语的概率论与数理统计。

given as Table 1.1

Table 1.1

Y 1 2 3

1 0.05 0.05 2 0.05 0.1 0.35

3 0 0.2 0.1

(a) Evaluate the marginal distribution of X and Y

(b) Determine whether the two random variables of X and Y are dependent or independent.

23.(a) IfX~N( 2，0.42), then findE(X 3)2

(b) Let (X,Y) be 2-dimensional random variables,and

(X,Y)(1,0)(1,1)(2,0)(2,1) P0.40.2ab

If E(XY) 0.8，find Cov(X,Y).

24.Suppose the probability distribution of Xi is as follows,andP(X1X2 0) 1. Find the correlation coefficient of X and Y.

25. Let X has a be uniform distribution on the interval (0,1),and Y X2 4X 1 Find (a)fY(y) (b) E(Y)

26. The random variable X, for fixed 0<p<1 and n≥1,has probability mass function (pmf)

cpkk 1,2,...,n px(x) otherwise 0

(a)Find c in terms of p and n.

(b)Find the cumulative distribution function of X.

27. If X~U[0,1]，Y~U[0,1],X and Y are independent. Find E|X Y|.

28. The length of time, in minutes, for an airplane to obtain clearance for take off at a certain airport is a random variable Y=3X-2, where X has the density function

1 x/4,x 0 ef(x) 4 elsewh.ere 0,

Find the mean and variance of the random variable Y.

适合于学习双语的概率论与数理统计。