Introduction to Probability and Statistics for Engineers and Scientists

4.5Properties of the Expected Value 117

where

Xi=

{

1 if theith letter is placed in its proper envelope

0 otherwise

Now, since theith letter is equally likely to be put in any of theNenvelopes, it follows
that

P{Xi= 1 }=P{ith letter is in its proper envelope}=1/N

and so

E[Xi]= 1 P{Xi= 1 }+ 0 P{Xi= 0 }=1/N

Hence, from Equation 4.5.2 we obtain that

E[X]=E[X 1 ]+···+E[XN]=

(

1

N

)

N= 1

Hence, no matter how many letters there are, on the average, exactly one of the letters will
be in its own envelope. ■

EXAMPLE 4.5g Suppose there are 20 different types of coupons and suppose that each time
one obtains a coupon it is equally likely to be any one of the types. Compute the expected
number of different types that are contained in a set for 10 coupons.

SOLUTION LetXdenote the number of different types in the set of 10 coupons. We
computeE[X]by using the representation

X=X 1 +···+X 20

where

Xi=

{

1 if at least one typeicoupon is contained in the set of 10

0 otherwise

Now

E[Xi]=P{Xi= 1 }

=P{at least one typeicoupon is in the set of 10}

= 1 −P{no typeicoupons are contained in the set of 10}

= 1 −

( 19

20

) 10