Let N denote a nonnegative integer-valued random variable. Show that k-1 k O In general show...
1. If X is a nonnegative integer valued random variable, show that n-1 n-0 Hint: Define the sequence of random variables I, n 1, by 1, if n X 10, ifn>X Now express X in terms of the I
1.9 Let Xi, -.. .Xn be nonnegative integer-valued random variables with identical pffx (-). A discrete mixture distribution W is created with pf fw (x)-puxi(x) +..+pfx, (x), where pi0 for i -1,... .n and X\-iPi1. Another random variable Y is defined by Y - (a) Compare the mean of W and Y. (b) If Xi,.. ,Xn are independent, compare the variance of W and Y.
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s 1 by is called the probability generating function of X (a) Show that d* (b) With 0 being considered even, show that PX is even) = P(-1) + P(1) (c) If X is binomial with parameters n and p, show that Pix is even) -1+12p (d) If X is Poisson with mean A, show that 1+ e-24 2 P[X is even)- (e) If X...
Problem 3.
3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It is flipped until two consecutive heads or two consecutive tails occur. Find the expected number of flips 5. Suppose that PX a)p, P[Xb-p, a b. Show that (X-b)/(a-b) is a Bernoulli variable, and find its variance
3. For a nonnegative integer-valued random variable X show that i-0 4. A coin comes up heads with probability p. It...
Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a) Prove that for any nonnegative integer m 7m k=0 b) Suppose that X~ B (n, p) and Y ~ B(m. p), and X, Y are independent. What is the distribution of the random variable Z X + Y? (c) Prove the following formula for binomial coefficients: n\ _n + m for kmin (m, n) (d) Let X ~ B (n, 1/2). What is P...
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
3. Suppose that X is a nonegative integer valued random variable. Show that E[X] = P(X ). Hint: Start with the formula EX= k= k, Now try to rearrange the terms. P(X = k) and for all positive integers k write
Q1. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function F and density f. Let b>0. (a) Write the forinula for E(X b)+1. (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1).
1, and let σ be a permutation of {1, , n). Recall...