# Expectation of the minimum of n independent Exponential Random Variables

Let $$X_1$$, $$X_2$$, · · · , $$X_n$$ be n independent Exponential random variables with mean 1. Find an expression for E(min($$X_1$$, $$X_2$$, · · · , $$X_n$$ )).

I have been studying random variables for a while but I don't know how to approach this one. Any help is appreciated!

$$P(min(X_1,..,X_n)>t)=P(X_1>t)^{n}=(e^{-t})^{n}=e^{-nt}$$. So the density function of $$min(X_1,..,X_n)$$ is $$ne^{-nt}$$ (for $$t>0$$). Can you find the expectation from this?
Let $$Y=\min(X_1,...,X_n)$$.
$$\mathbb P\{Y\geq y\}=\mathbb P\{X_1\geq y,...,X_n\geq y\}=\mathbb P\{X_1\geq y\}^n,$$ where the last equality come from independence. The density function is therefore given by $$f_Y(y)=-n\lambda e^{-\lambda x}(e^{-\lambda x}-1)^{n-1}\cdot \boldsymbol 1_{[0,\infty )}(x).$$