# Question: If X and Y are independent binomial random variables with

If X and Y are independent binomial random variables with identical parameters n and p, show analytically that the conditional distribution of X given that X + Y = m is the hypergeometric distribution. Also, give a second argument that yields the same result without any computations.

Suppose that 2n coins are flipped. Let X denote the number of heads in the first n flips and Y the number in the second n flips. Argue that given a total of m heads, the number of heads in the first n flips has the same distribution as the number of white balls selected when a sample of size m is chosen from n white and n black balls.

Suppose that 2n coins are flipped. Let X denote the number of heads in the first n flips and Y the number in the second n flips. Argue that given a total of m heads, the number of heads in the first n flips has the same distribution as the number of white balls selected when a sample of size m is chosen from n white and n black balls.

## Answer to relevant Questions

Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X1 + X2 and Y = X2 + X3. The random vector X, Y is said to have a bivariate Poisson distribution. Find its ...Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, β). That is, its density is f(w) = βe−βw(βw)t−1 / Γ(t), w > 0. Suppose also that given that W = w, the ...Verify Equation (6.6), which gives the joint density of X(i) and X(j). Let X and Y be independent continuous random variables with respective hazard rate functions λX(t) and λY(t), and set W = min(X,Y). (a) Determine the distribution function of W in terms of those of X and Y. (b) Show that ...Suppose that Y is a normal random variable with mean μ and variance σ2, and suppose also that the conditional distribution of X, given that Y = y, is normal with mean y and variance 1. (a) Argue that the joint distribution ...Post your question