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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 ?

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