Introduction to Probability and Statistics for Engineers and Scientists

136 Chapter 4:Random Variables and Expectation

cis the median of the distribution function ofX. Prove this result whenX is

continuous with distribution functionFand density functionf.(Hint: Write

E[|X−c|] =

∫∞

−∞

|x−c|f(x)dx

=

∫c

−∞

|x−c|f(x)dx+

∫∞

c

|x−c|f(x)dx

=

∫c

−∞

(c−x)f(x)dx+

∫∞

c

(x−c)f(x)dx

=cF(c)−

∫c

−∞

xf(x)dx+

∫∞

c

xf(x)dx−c[ 1 −F(c)]

Now, use calculus to find the minimizing value ofc.)

36.We say thatmpis the100p percentileof the distribution functionFif

F(mp)=p

Findmpfor the distribution having density function

f(x)= 2 e−^2 x, x≥ 0

37.A community consists of 100 married couples. If during a given year 50 of the

members of the community die, what is the expected number of marriages that

remain intact? Assume that the set of people who die is equally likely to be any of

the

(

200

50

)

groups of size 50. (Hint: Fori=1,..., 100 let

Xi=

{

1 if neither member of coupleidies

0 otherwise

38.Compute the expectation and variance of the number of successes innindepen-

dent trials, each of which results in a success with probabilityp. Is independence

necessary?

39.Suppose thatXis equally likely to take on any of the values 1, 2, 3, 4. Compute

(a)E[X]and(b)Var(X).

40.Letpi=P{X=i}and suppose thatp 1 +p 2 +p 3 =1. IfE[X]=2, what values

ofp 1 ,p 2 ,p 3 (a)maximize and(b)minimize Var(X)?

41.Compute the mean and variance of the number of heads that appear in 3 flips of

a fair coin.