Let X denote the amount of time a book on twohour reserve is

Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following.

F(x) =

Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.)

(g)    V(X) and ?_x

(h) If the borrower is charged an amount h(X) = X² when checkout duration is X, compute the expected charge

E[h(X)].

Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = ?x = (h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)]. 0 x

Solution

Given :

F(x)= 0 , x < 0
x^2/16 , 0 ? x ? 4
1 , 4 ? x

Solution :

g)
E(x^2) = (2/16) ? x^2 x dx --- integrate from 0 to 4
(2/16) ? x^3 dx --- integrate from 0 to 4
= 2x^4/64 from 0 to 4
= x^4/32 from 0 to 4
=8

E(x) = (2/16) ? x x dx --- integrate from 0 to 4
= (2/16) ? x^2 dx --- integrate from 0 to 4
= (2/16) (x^3/3)
= (2/48) x^3 from 0 to 4
= (2/48) [ 4^3 - 0^3] = 8/3

V(x) = E(x^2)- [E(x)]^2 = 8 - (8/3)^2
= 8/9= 0.8889
SD(x) = sqrt (0.8889) = 0.9428 Answer

h)
E(h(x)) = E(x^2) --- this has already been computed at the beginning of g)
E(x^2) = 8 Answer