Question

Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given X-k, for some k-n. Let a = P(X-k), and suppose that the value of a has been computed (a) Give the inverse transform method for generating Y. (b) Give a second method for generating Y (c) For what values of a, small or large, would the algorithm in (b) be inefficient?

Answer 1

Let Xbe a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Ywhoseprobability mass function is the same as the conditional mass function of X given that X2k, for some kn. Let α = P [X 2 kl and suppose that the value of α has been computed. a) Give the inverse transform methodfor generating Y. (x k) P(X=i) 0 if n izk Otherwise And this is n! Otherwise

n-l p From the recursiveidentity: P(Y-1FP(Y-i) With idenoting the value currently under consideration, pr- P(r-i)the probability that Yis equal to i,and F -F (i)the probability that Yis less than or equal to i. Algorithm: i+11-p Step 1: Generate a random number U Step 2: c = p/(1-2), i = k, pr = aRTHypk (1-2)"-k, F = pr Step 3: If (U < F), set Y-i and stop. Step 4: pr = c(n-i)pr/(i + 1), F = F + pr, i = i + 1 Step 5: Go to step 3 b) Given a second methodfor generating Ywe may use Acceptance-Rejection dominating these probabilities using a binomial distribution. Let Z be a random variable having a Binomial (n, p)distribution, with probability mass q, then let c be a constant such that:

for allj such thatp >0 ; Letc Ij Algorithm: Step-1: Simulate the value of Z. Step-2: Generate a random number U Step-3 : If (Ụ < P2 | qz), set y-z and stop. Otherwise return to step-1. For small values of α the rejection method in (b) will be inefficient.