The idea used in Exercise 318 generalizes to give a new form

The idea used in Exercise 3.18 generalizes to give a new formula for the expected value of any nonnegative integer valued random variable. Show that if the random variable X only takes nonnegative integers as its values then E(X) = summation k = 1 infinity P(X > k). This holds even when E(X) = infinity, in which case the sum on the right hand side is infinite. Write P(X > k) as P(X = i) in the sum, and then switch the order of the two summations.

Solution

E(X) = sigma ( 1 to infinity) P(X>=k)

=> E(X) = sigma ( k= 1 to infinity) sigma ( i =k to infinity ) P( X=i )

=> E(X) = sigma ( k= 1 to infinity) { P( X=k)+P( X=k+1)+P( X=k+2)+.......+P( X=infinity) }

=> E(X) = { P( X=1)+P( X=2)+P( X=3)+.......+P( X=infinity) } +

                { P( X=2)+P( X=3)+P( X=4)+.......+P( X=infinity) } +

                { P( X=3)+P( X=4)+P( X=5)+.......+P( X=infinity) } +

               ......   { P( X=infinity)+P( X=infinity)+P( X=infinity)+.......+P( X=infinity) }

= { P( X=1) + 2 P( X=2) + 3P( X=3)+ .......+ infinity * P( X=infinity) }

Since , given that all the values are non negative,

E(X) = Positive and infinity because this parameter { infinity * P( X=infinity)} will reach infinity.

Hence E(X) = infinity is true

Hence given euation holds.