Let X and Y be two exponentially distributed and independent random variables with mean alpha and beta respectively. If Z = min(X, Y), then the mean of Z is given by A. 1/(alpha + beta) B. min(alpha, beta) C. (alpha * beta)/(alpha + beta) D. (alpha + beta)/2

GATE 2004 · Engineering Mathematics · Exponential Distribution · medium

Answer: (alpha * beta)/(alpha + beta)

1. Convert means to rates: lambda_X = 1/alpha, lambda_Y = 1/beta.
2. Rate of the minimum: min(X,Y) is exponential with rate (alpha + beta)/(alpha * beta).
3. Mean of Z: E[Z] = 1 / [(alpha + beta)/(alpha * beta)] = (alpha * beta)/(alpha + beta).