Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y...
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...
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...
10) (11) Let X and Y be 2 independent random variables. Suppose X ~ Gamma(0, 38) and Y ~ Gamma(a, 2B). Let 2 = 2X +3Y. Determine the probability distribution of Z. (Hint: use the method of moment-generating functions
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x =-2,-1,0 p(y) = 1/2 for y =1,6 Z = X + Y. What is the distribution of Z using the method of MGF's
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
Suppose U and V are independent geometric random variables with parameter p. Let Z = U + V . Determine the conditional probability mass function of pU|Z(·| n) of U given that Z = n.
Prove that Box-Muller method described in class generates independent standard normal random variables. 4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample...