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Chapter 5: Q 5.8 (page 215)

Let Xbe a random variable that takes on values between0andc. That isP0鈮�X鈮�c=1.Show that

Var(X)鈮�c24

Hint: One approach is to first argue that

localid="1646883602992" EX2鈮�cEX

and then use this inequality to show that

Var(X)鈮�c2[伪(1-伪)]where伪=E[X]c

Short Answer

Expert verified

Hence, proved thatVar(X)鈮�c24.

Step by step solution

01

Given information:

LetXbe a random variable that takes on values between0andc. That is,P0鈮�X鈮�c=1

Hint: One approach is to first argue that

E[X2]鈮�cE[X]

and then use this inequality to show that

Var(X)鈮�c2[伪(1-伪)]where伪=E[X]c

02

Explanation:

SinceXbe, a random variable that takes on values between 0andc, we know that

f(x)=0ifx鈮�0orx>0

Thus,

E[X2]=鈭�0cx2f(x)dx鈮�鈭�0ccxf(x)dx=cE[X]

03

Explanation:

Hence,

Var(X)=EX2-EX2鈮�cEX-EX2=c2(伪-伪2)

Where伪=E[X]cwe know伪-伪2attains its maximum value of14at伪=12so

VarX鈮�c2(伪-伪2)伪=12=c212-14=c24

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Most popular questions from this chapter

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameters位=12. What is

(a)the probability that a repair time exceeds2hours?

(b)the conditional probability that a repair takes at least10hours, given that its duration exceeds9hours?

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.

(a) Show that E[g'(Z)]=E[Zg(Z)];

(b) Show that E[Zn+1]=nE[Zn-1].

(c) Find E[Z4].

Consider the beta distribution with parameters (a,b). Show that

(a) when a>1and b>1, the density is unimodal (that is, it has a unique mode) with mode equal to (a-1)/(a+b-2)

(b) when a鈮�1, b鈮�1, and a+b<2, the density is either unimodal with mode at 0or 1or U-shaped with modes at both0and1;

(c) when a=1=b, all points in [0,1]are modes.

Let Xbe a random variable with probability density function

f(x)=c(1-x2)鈭�1<x<10otherwise

(a) What is the value of c?

(b) What is the cumulative distribution function of X?

The lung cancer hazard rate位(t)of a t-year-old male smoker is such that

位(t)=.027+.00025(t鈭�40)2t鈮�40

Assuming that a 40-year-old male smoker survives all other hazards, what is the probability that he survives to

(a)age50and (b)age 60 without contracting lung cancer?
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