Solution
= Solution
For $t\geq0$, <independence> factors the <survival function> of the minimum:
$$
\mathbb P(M>t)=\prod_{j=1}^n\mathbb P(X_j>t)
=\prod_{j=1}^ne^{-\lambda t}=e^{-n\lambda t}.
$$
The minimum is nonnegative, and this is exactly the <survival function> of an <exponential distribution>. Hence
$$
\boxed{M\sim\operatorname{Exp}(n\lambda).}
$$
The parameter here is the rate, rather than the mean. The calculation is the simplest example of <competing exponential clocks>.