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Chapter 7: Q.29 (page 361)

Let X1,鈥�,Xnbe independent and identically distributed random variables. Find

EX1鈭�X1+鈰�+Xn=x

Short Answer

Expert verified

EX1鈭�X1+鈰�+Xn=x=xn

Step by step solution

01

Given information

Given in the question that, letX1,鈥�,Xnbe independent and identically distributed random variables.

02

Explanation

Given,

and since all the variables are equally distributed, we have the symmetry

for allj=1,鈥�,n. Using the linearity of the conditional expectation, we have that

=鈭�j鈥�EXj鈭�X1+鈰�+Xn=x

=鈭�j鈥�EX1鈭�X1+鈰�+Xn=x

=nEX1鈭�X1+鈰�+Xn=x

Which implies that,

EX1鈭�X1+鈰�+Xn=x=xn

03

Final answer

EX1鈭�X1+鈰�+Xn=x=xn

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