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Chapter 8: Q.8.3 (page 393)

If E[X]=75E[Y]=75Var(X)=10

Var(Y)=12Cov(X,Y)=-3

give an upper bound to

(a) P{|X-Y|>15};

(b) P{X>Y+15};

(c)P{Y>X+15};

Short Answer

Expert verified

An upper bound to

(a). P{|X-Y|鈮�15}鈮�.1244

(b).P{X>Y+15}鈮�.1107

(c).P{Y>X+15}鈮�.1107

Step by step solution

01

Part (a) Step 1: Given Information

Let Xand Ybe radnom variables such that

E[X]=75,E[Y]=75,Var(X)=10Var(Y)=12,Cov(X,Y)=-3
02

Part (a) Step 2: Explanation

Consider the random variable

A=X-Y

Since

渭A=E[A]=E(X)-E(Y)=0

and

蟽A2=Var(A)=Var(X)+Var(Y)-2Cov(X,Y)=28

using Chebyshev's inequality we get

P{|X-Y|鈮�15}=PA-渭A鈮�15鈮�蟽A2152=28152=.1244

03

Part (b) Step 1: Given Information

Let Xand Ybe random variables such that

localid="1650024398003" E[X]=75,E[Y]=75,Var(X)=10Var(Y)=12,Cov(X,Y)=-3

04

Part (b) Step 2: Explanation

Again, consider the random variable A. Now, using Proposition 5.l we get

P{X>Y+15}=P{A>15}鈮�蟽A2蟽A2+152=2828+152=.1107

05

Part (c) Step 1: Given Information

Let Xand Ybe radnom variables such that

E[X]=75,E[Y]=75,Var(X)=10Var(Y)=12,Cov(X,Y)=-3
06

Part (c) Step 8: Explanation

If we denote as B=Y-X, then this random variable has mean

渭B=E[B]=E(Y)-E(X)=0

and variance

蟽B2=Var(B)=Var(Y)+Var(X)-2Cov(X,Y)=28

Using One-sided Chebyshev inequality we get

P{Y>X+15}=P{B>15}鈮�蟽B2蟽B2+152=2828+152=.1107.

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