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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
- \( E( R)\)
- the joint density of the min and the max of the \(U\)’s
- \(P( R>0.5)\)
[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
- \(\mathbf{E} R\)
- the joint density of the max and the min of the \(U_i\)’s.
- 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 ?