Question

Random variables $X_i\sim exp(\lambda _{i})$ are independent of each other, where i=1,2,3 and that $\lambda _{i}>0,\forall i$

Calculate P(X1<X2<X3)

Answer 1

We know that,

Min(X1, X2,, ...., Xn) ~ Exp($\lambda_1 &plus; \lambda_2 &plus; ...&plus; \lambda_n$)

and,

P(Xi = Min(X1, X2, ...., Xn)) = $\lambda_i / (\lambda_1 &plus; \lambda_2 &plus; ...&plus; \lambda_n)$

and

P(Xi < Xj) = $\lambda_i / (\lambda_i &plus; \lambda_j)$

Now,

P(X1<X2<X3) = P(X1 = min(X1,X2,X3))· P(X2 < X3|X1 = min(X1,X2,X3))

By memoryless property of exponential distribution,

P(X2 < X3|X1 = min(X1,X2,X3) = P(X2 < X3)

Thus,

P(X1<X2<X3) = P(X1 = min(X1,X2,X3))· P(X2 < X3)

$= \frac{\lambda_1}{\lambda_1 &plus;\lambda_2&plus; \lambda_3} .\frac{\lambda_2}{\lambda_2&plus; \lambda_3}$