Let $X_i, i = 1, 2, 3,$ be independent exponential random variables with rates $\lambda_i, i = 1,2,3.$
How does one derive the following:
$$\mathbb P \{\min(X_1, X_2, X_3) = X_1\} = \frac{\lambda_1}{\lambda_1 +\lambda_2 + \lambda_3}?$$
I see this used all of the time, and I'm familiar with the fact that
$$\mathbb P \{X_1 < X_2\} = \frac{\lambda_1}{\lambda_2 +\lambda_2},$$
which I assume is property one uses to get from the latter to the former; but, I've been working with the definitions, and whatnot, with no luck. Any help here would be appreciated. Thanks!
Best Answer
Observe that $P(\min(X_1,X_2,X_3)=X_1) = P( \min (X_2,X_3)> X_1)$. Since $\min (X_2,X_3)$ is exponential with parameter $\lambda_2+\lambda_3$, and is also independent of $X_1$, the result follows from the stated formula for the minimum of two independent exponential random variables.