Question

1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable. Express the probability Pf_2? X 3c) in terms of values ?(t) for t 0. 3) Let X1, X2,..., Xn,... be a sequence of i.i.d. RVs with the mean equal to 0 and the variance equal to 1. Form the corresponding sequence of Mn (Xi+... +Xn)/n. How large should n be to guarantee that the probability of Mn being at least 2 units away from 0 is at most 1/100? 4) Let Y be the standard normal random variable. Let X = 10Y + 10, Compute the mean and the variance of X 5) Let X be exponential random variable with parameter 1/2, and let Y be Poisson random variable with mean 2. b. Use Markov's inequality to estimate P(Y25) c. Use Chebyshev's inequality to estimate P(X 2 5). d. Use Chebyshev's inequality to estimate P(Y 25

Answer 1

by Sun law) 2 E (x)-F(X) V(x)-V(7) AN (O,