Test
1. Consider the set ???1,2,3,4,5,6,7,8,9? with subsets A??1,3,5,7,9?,B??2,4,6,8?,C??1,2,3,4?,D??7,8?
Find the following sets:
(a) A?D and A?B (b) (C?D)c and A?(B?D)
ccc2. ShowDe Morgans Law: (a) (E?F)?E?F,(b)(E?F)?E?F
ccc3.(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(x) 0.2 0.16 0.41 0.23
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)??2x?1e?2??2x?0
x?0Let Y?X, Find E(Y)
9.Let X be a continuous random variable with probability density function
?xe?xx?0f(x)??otherwise?0(a) Find P(1 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)?，P(A|B)?1523，P(B|A)?.FindP(B). 3515.(a) Suppose A, B are two events, and P(A)=1/4, P(B)=1/2, P(AB)=1/9, then evaluate P?AB? (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) 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. 19.(a)The random variable X has the probability mass function below: X 1 2 3 4 Px(x) 0.4 Find a and FX(3.2) (b) A continuous random variable X having the probability density function ?x2?f(x)??3?0? 0.14 a 0.24 ?1?x?2elsewhere.Find ???f(x)dx and P(0 ??20.(a)Let P(BA)?P(BA),P(A)?(b) Suppose X and Y 1?P(B).FindP(AB) 3are independent random variables
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