Sum of exponentials, distrubution

Question

The sum of $n$ number of independent $X_i$ where each $X_i$ follows $$\exp(\lambda_i),$$ then $$Y=\min [ X_1, \dots , X_n]$$ follows ... what distrubution? Is it $exp( \Sigma \lambda_i)$?

Answer 1

Hint: For $y\in\mathbb{R}$: $$ P(Y\geq y)=(X_1\geq y,\ldots,X_n\geq y). $$ Without assuming anything about the joint distribution of $(X_1,\ldots,X_n)$ you can't say more. But if you're willing to assume that $X_1,\ldots,X_n$ are independent, then the probability above factors.