# Category Archives: Order Statistics

## Uniform Spacing

Let $$U_1, U_2, U_3, U_4, U,5$$ be independent uniform $$(0,1)$$ random variables. Let $$R$$ be the difference between the max and the min of the random variables. Find

1. $$E( R)$$
2. the joint density of the min and the max of the $$U$$’s
3. $$P( R>0.5)$$

[Pitman p. 355 #14]

## Difference between max and min

Let $$U_1,U_2,U_3,U_4,U_5$$ be independent, each with uiform distribution on $$(0,1)$$. Let $$R$$ be the distance between the max and the min of the $$U_i$$’s. Find

1. $$\mathbf{E} R$$
2. the joint density of the max and the min of the $$U_i$$’s.
3. the $$\mathbf{P}(R> .5)$$

[pitman p355, #14]

## Order statistics II

Suppose $$X_1, … , X_{17}$$ are iid uniform on $$(.5,.8)$$. What is $${\mathbf{E}} [X_{(k)}]$$ ?

## Order statistics I

Suppose $$X_1, … , X_n \stackrel{iid}{\sim} U(0,1)$$. How large must $$n$$ be to have that $${\mathbf{P}}(X_{(n)} \geq .95) \geq 1/2$$ ?

## Joint of min and max

Let $$X_1,…,X_n \stackrel{iid}{\sim} \mbox{Exp}(\lambda)$$

Let $$V = \mbox{min}(X_1,…,X_n)$$ and  $$W = \mbox{max}(X_1,…,X_n)$$.

What is the joint distribution of $$V,W$$. Are they independent ?