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Chapter 7: Q.7.30 (page 361)

Consider Example 4f, which is concerned with the multinomial distribution. Use conditional expectation to compute E[NiNj], and then use this to verify the formula for Cov(Ni, Nj) given in Example 4f.

Short Answer

Expert verified

Therefore it has been computed that ENiNj=m(m鈭�1)pipjby using conditional expectation

Step by step solution

01

Given information

Use conditional expectation to compute E[NiNj],

02

Solution

Compute the ENiNjusing conditional expectation.

So,

ENiNj鈭�Ni=NiENj鈭�Ni

=Nim鈭�Nipj1鈭�pi

Now, each of the m-Ni trails no resulting in outcome i will independently result inj with probability pj1-pi.

03

Solution

Therefore,

ENiNj=pj1鈭�pimENi鈭�ENi2

=pj1鈭�pim2pi鈭�m2pi2鈭�mpi1鈭�pi

=pj1鈭�pim(m鈭�1)pi1鈭�pi

=m(m鈭�1)pipj

And,

Cov鈦�Ni,Nj=m(m鈭�1)pipj鈭�m2pipj

=m2pipj鈭�mpipj鈭�m2pipj

=鈭�mpipj

04

Final answer 

Therefore it has been computed that ENiNj=m(m鈭�1)pipjby using conditional expectation

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Most popular questions from this chapter

If Y=aX+bwhere a and b are constants, express the moment generating function of Y in terms of the moment generating function of X.

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Show how to compute Cov(X,Y) from the joint moment generating function ofX and Y.
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