Minimum and average of uniform draws

Topic: Joint distributions · more questions on this topic
A joint table ›

Numbers X_1,\ldots,X_n are drawn independently from the uniform distribution on [0,1]. Let M=\min(X_1,\ldots,X_n) and let \overline X=(X_1+\cdots+X_n)/n.

1. Express M and \overline X in terms of X_1,\ldots,X_n.

2. Find the density of M.

3. Find \mathbb{E}\overline X and \operatorname{Var}(\overline X).

Worked answer (click to show)

1. M=\min(X_1,\ldots,X_n) and \overline X=(X_1+\cdots+X_n)/n.

2. For 0\le x\le 1, independence gives \mathbb{P}(M>x)=(1-x)^n. The distribution function is 1-(1-x)^n on that interval, and the density is n(1-x)^{n-1}.

3. \mathbb{E}\overline X=\mathbb{E} X_1=1/2. Independence gives \operatorname{Var}(\overline X)=n^{-1}\operatorname{Var}(X_1)=1/(12n).