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Chapter 7: Q.26 (page 361)

Prove thatE[g(X)Y鈭�X]=g(X)E[Y鈭�X].

Short Answer

Expert verified

We prove that,E[g(X)Y鈭�X]=g(X)E[Y鈭�X]

Step by step solution

01

Given information

Given in the question that, we have to prove thatE[g(X)Y鈭�X]=g(X)E[Y鈭�X]

02

Explanation

Because of the clarity, suppose that Xand Yare continuous random variables with its density functions. Take any x鈭�suppfX. We have that

E(g(X)Y鈭�X=x)=鈭�鈩�2鈥�g(x)yf(X,Y)鈭�X=x(x,y)dxdy

But, we have that

f(X,Y)鈭�X=x(x,y)dxdy=fY鈭�X(y鈭�x)dy

So the integral becomes

鈭�鈩�鈥�g(x)yfY鈭�X(y鈭�x)dy

Which is equal to

g(X)E(Y鈭�X=x)

Since the relation hold for every x鈭�suppf, we end up with

E(g(X)Y鈭�X)=g(X)E(Y鈭�X)

03

Final answer

We prove that,E[g(X)Y鈭�X]=g(X)E[Y鈭�X]

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Most popular questions from this chapter

A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability

Pi,i=1,鈥�,r鈭�1rPi=1

(a) Compute the mean number of insects that are caught before the 铿乺st type 1catch.

(b) Compute the mean number of types of insects that are caught before the 铿乺st type1 catch.

If E[X]=1and Var(X)=5find

(a)E[(2+X2)]

(b)Var(4+3X)

The number of winter storms in a good year is a Poisson random variable with a mean of 3, whereas the number in a bad year is a Poisson random variable with a mean of5. If next year will be a good year with probability .4or a bad year with probability .6, find the expected value and variance of the number of storms that will occur.

Show that E[(X鈭�a)2] is minimized at a=E[X].

Let X be the length of the initial run in a random ordering of n ones and m zeros. That is, if the first k values are the same (either all ones or all zeros), then X 脷 k. Find E[X].
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