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Chapter 6: Q. 6.2 (page 277)

The joint probability mass function of the random variables X, Y, Z is

p(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=14

Find (a) E[XYZ], and (b) E[XY + XZ + YZ].

Short Answer

Expert verified

(a) EXYZ=6

(b)EXY+XZ+YZ=10

Step by step solution

01

Given information (part a)

The joint probability mass function of the random variables X, Y, Z is p(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=14

E[X]=鈭�xP(X=x)where X denotes a random variable.

02

Explanation (part a)

The random variable XYZ can assume the following values:

1脳2脳3=62脳1脳1=22脳2脳1=42脳3脳2=12

Use the formula: E[X]=鈭�xP(X=x)

Also,

p(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=14

So,

E[XYZ]=14(6)+14(2)+14(4)+14(12)

=244

=6

03

Given information (part b)

The random variable XY+XZ+YZcan assume the following values:

1脳2+2脳3+1脳3=2+6+3=112脳1+1脳1+2脳1=2+1+2=52脳2+2脳1+2脳1=4+2+2=82脳3+2脳2+3脳2=6+4+6=16

Use the formula:E[X]=鈭�xP(X=x)

04

Explanation (part b)

p(1,2,3)=p(2,1,1)=p(2,2,1)=p(2,3,2)=14

So,

E[XY+YZ+XZ]=14(11)+14(5)+14(8)+14(16)

=11+5+8+164=404=10

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Most popular questions from this chapter

Let X1,X2,...be a sequence of independent uniform (0,1)random variables. For a fixed constant c, define the random variable N by N=min{n:Xn>c}Is N independent ofXN? That is, does knowing the value of the first random variable that is greater than c affect the probability distribution of when this random variable occurs? Give an intuitive explanation for your answer.

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