概率论与数理统计英文题目

1、精选优质文档-倾情为你奉上Test1. Consider the set with subsets Find the following sets: (a) and (b) and 2. ShowDe Morgans Law: (a) ,(b)3.(a)Let P(A)=0.5,P(B)=0.4,P( AB)=0.2. Find P(AB) 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

2、 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. . Show that (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. . Find t

3、he correlation coefficient of X and Y.6. (a)Let X Uniformly distributed on (-1,1), i.e. XU(-1,1). Find the Expectation and Variance of X.(b)Let X and Y be continuous random variables with joint pdf Let Z=X+Y, Find. In particular, if X and Y are independent, Find fX*fY.7.(a)The random variable X has

4、pmf is: X 0 1 2 3 PX(x) 0.2 0.16 0.41 0.23 Find P(1X<3) and FX(1.7).(b)Let X be a continuous random variable with pdf Find P(X=3), P(X>1) and.8. X is continuous r.v. with pdf Let , Find E(Y)9.Let X be a continuous random variable with probability density function (a) Find P(1<X<2) (b) Fi

5、nd: (1) E(X); ,(2) Given that E(X2)=6. Find Var(X)10If the joint probability density function of two random variables is given by (a) Find the marginal pdf of X and Y. (b)Determine whether the two random variables are dependent or independent.11Suppose the random variable X has the density function

6、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

7、 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 (b)Let，.Find.15.(a) Suppose A, B are two events, and P(A)=1/4, P(B)=1/2, P(AB)=1/9, then evaluate (b)Use Bayes Theorem to show that if P(A),

8、P(B)>0 and then 16.(a)Let X has a Binomial distribution with parameters n and p, i.e. X b (n, p).Find (b) Show that 17. Suppose the distribution function of a random variable X is，Find (a) the probability that X gets value within (0.3，0.7);(b) the density function of X18. The operational lifetime

9、 X, in years, of a battery powered watch has probability density function (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.19.(a)The random variable X has the probability mass function

10、below:X1234Px(x)0.40.14a0.24Find a and (b) A continuous random variable X having the probability density function .Find and P(0<X<1).20.(a)Let .FindP(AB)(b) Suppose X and Y are independent random variables ,and Find21. Suppose the density function of (X,Y) is (a) Determine the constant k. (b)

11、Find the probability PX<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 given as Table 1.1Tabl

12、e 1.1 X Y 1 2 3 1 0.05 0.05 0.1 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) If, then find(b) Let (X,Y) be 2-dimensional random variables,and If ，find .24.Suppose the probab

13、ility distribution of is as follows,and.Find the correlation coefficient of X and Y.25. Let X has a be uniform distribution on the interval (0,1),and Find (a) (b) E(Y)26. The random variable X, for fixed 0<p<1 and n1,has probability mass function (pmf) (a)Find c in terms of p and n.(b)Find the

14、 cumulative distribution function of X.27. If ，,X and Y are independent. Find .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 Find the mean and variance of the random variable Y.29. Suppose the probability density function of random variable X is ,and .Find (a) the probability density function of Y,.(b) expectation E(Y) and variance Var(Y).30. Suppose X and