Lent 2017 MATHEMATICAL TRIPOS: PART IA PROBABILITY Example Sheet 3 (of 4) JRN 1. Let x1, X2, ... , In be positive real numbers. (a) Show that their harmonic mean is no greater than their arithmetic mean, that is, -1 n 1 1 ? n n - n > i=1 1 i=1 (b) Show that, if y1, ... , Un is any reordering of x1, ... , In, then 1 i=1 n yi > 1. n 2. Let X be a random variable. (a) Show that, for all p € (0, 00) and all x € (0, 00), P(|X| ?x) ? E(|X|P)x P. (b) Show that, for all 3 ? 0, P(X ? x) ? E(efX)e-Bx 3. Let X be a Poisson random variable of parameter À € (0, 00). (a) By optimizing the estimate of Question 2(b) over 6, show that, for all x ? A, P(X ? x) ? exp{-x log(x/>) ->+x}. (b) Show, on the other hand, that, for integers x, as x >00, P(X =x) ~. 1 ? exp{-x log(x/>) ->+x}. 4. Consider a random sample X1, ... , Xn taken from a distribution having mean u and variance o2 < 0. Use Chebyshev's inequality to determine a sample size n that will be sufficient, whatever the distribution, for the probability to be at least 0.99 that the sample mean X will be within two standard deviations of p. 1
5. Suppose we conduct a sequence of independent Bernoulli trials and denote by X the number of trials up to and including the ath success. Show that P(X = r) = a-1) r - 1 ) p " q " - " , r = a, a + 1, .... Show that the generating function for this distribution is pata(1-qt)-a. Deduce that E(X) = a/p and var(X) = aq/p2. Explain how X can be represented as the sum of a independent random variables, all with the same distribution. Use this representation to derive again the mean and variance of X. 6. For a random variable X with mean u and variance o2 < , define the function V(x) = E((X-x)2). Express the random variable V(X) in terms of u, 02 and X, and hence show that E(V(X)) = 202. 7. Suppose X is a real-valued random variable and f : R -> R and g : R -> R are two non-decreasing functions. Prove the 'Chebyshev order inequality': E(f(X))E(g(X))?E(f(X)g(X)). Hint. Consider (f(X1) - f(X2))(g(X1) - g(X2)) where X1 and X2 are independent copies of X.] 8. Let (Xn : n E N) be a sequence of independent identically distributed random variables, with mean u and variance o2 < . Set So = 0 and Sn = X1 + ... + Xn for n > 1. Let N be a bounded non-negative integer-valued random variable which is independent of the sequence (Xn : n E N). (a) Show that E(SN) = pE(N). (b) Show that E(S?|N = n) = no2 + n2u and hence express the var(SN)