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1、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 numbered 1 t
2、hrough 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 the correlatio
3、n 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 pmf is: X 0 1
4、 2 3 PX(x) 0.2 0.16 0.41 0.23 Find P(1X1) 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(1X0 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. Supp
5、ose 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 X, in years, of a battery powered watch has probability density function (a) Find the value of c.(b) Find the cumulative distr
6、ibution 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 below:X1234Px(x)0.40.14a0.24Find a and (b) A continuous random variable X having the probability density function .Find and P(0
7、X1).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) Find the probability PX1,Y3.22. Let X denote the number of times a certain numerical control machine will malfunction: 1,2,or 3 times o
8、n 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 1.1Table 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 ran
9、dom 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 probability 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,
10、1),and Find (a) (b) E(Y)26. The random variable X, for fixed 0p1 and n1,has probability mass function (pmf) (a)Find c in terms of p and n.(b)Find the 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
11、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 Y are two random variables. The joint probability density function isFind (a) the marginal density of X.(b) expectation E(Y) and variance Var(Y).(c)Determine whether the two random variables are dependent or independent ？