Question

1) Suppose that E(Y∣X)=X^2. Then E(Y/X) is equal to which of the following?

a) 1 b) E(X) c) E(X^2) d) E(Y)

2)Var(Y∣X=x) is less than or equal to Var(Y) unless Var(Y)=0. True or False?

Answer 1

Solution :

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1) Suppose that E(Y∣X)=X^2, then we have to determine the value of E(Y/X) :

Thus, we have the following information,

Taking expectations on both sides , we get ,

  

  

  

  

But , we are given that ,

Thus , we can write that ,

Thus , finally , we can write ,

Hence , the correct answer is Option (B) --- E(X) ................................ (Ans)

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2) To find whether "Var(Y∣X=x) is less than or equal to Var(Y) unless Var(Y)=0" is True or False :

From the Law of Total Variance , we know that the variance of a random variable (say Y) is nothing but the sum of the expected conditional variance of Y given X (say) and the variance of the conditional expectation of Y given X . So , mathematically , we can write ,

Now , since variance of a random variable is always non-negative , so we can write ,

Thus, clearly, we can say that E(Var(Y|X=x)) is less than or equal to Var(Y) unless Var(Y)=0.

Thus , the statement provided is FALSE ......................................... (Ans)

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