Question

3. (8 pts.) Suppose that Xi ~ Exp(A) and X2 ~ Exp(A2) where λ1 and λ2 are positive con- λ2, but do assume that Xi and X2 are independent. Compute stants. Do not assume λι P(X1 < X2). Now note that the probability you just computed is in fact P(Xmin(XI, X2)). This suggests the following generalization. Suppose we have a collection of N independent ex- ponential random variables, X1, X2, , XN- Each Xi ~ Exp(Ai); again, do not assume anything about the At's except that they are all positive constants. Compute = min{ X1 , X2, . . . XN

This is a probability question. Please be thorough and detailed.

Answer 1

P(X1 < X2) =

$\int _0^{\infty }\int _0^{x_2}\:\:\:\lambda _1 \lambda _2e^{-\left(\lambda _1x_1+\lambda _2x_2\right)}\:dx_1\:dx_2 = \frac{\lambda _1}{\lambda _1+\lambda _2}$

similarly

P(X1 = min(X1,X2,..Xn))

=$\frac{\lambda _1}{\lambda _1+\lambda _2+...\lambda _n}$