For each of the conditions (a)-(d) below, give examples of random variables that satisfy the condition, and justify. If no such random variables exist, say so and provide justification.
a) Random variables X, Y such that Var (X + Y ) < Var X + Var Y
b) Random variables X, Y such that E(X|Y = y) > E(X) for all y
c) Independent random variables X, Y such that X+Y/Sqrt(2) has the same pdf
as X
d) Random variables Xi that are i.i.d, with mean and variance equal to 1,
such that P( Summation of Xi > 2n, for i=1 to n) tends to 0.5 as n tends
to become infinite
I have solved (a) and it would be true for negative covariance. How to solve the rest. Any help would be appreciated.
Best Answer
b) says $EXI_{Y=y} >EX P\{Y=y\}$ in the disecrete case. Summing over $y$ we get $EX>EX$ In general case use the fact that $E(EX|Y))=EX$ to get the contradiction $EX >EX$. So b) is not possible. For c) take $X,Y$ i.i.d. standard normal and use the fact that characteristic function of standard normal distribution is $e^{-t^{2}/2}$. Using this you can easily check that $\frac {X+Y} {\sqrt 2}$ also has the same characteristic function. [$Ee^{it\frac {X+Y} {\sqrt 2}}=Ee^{it\frac X {\sqrt 2}}Ee^{it\frac Y {\sqrt 2}}=e^{-t^{2}/4}e^{-t^{2}/4}=e^{-t^{2}/2}$].
d) is not possible by SLLN: since $\frac 1 n \sum_{i=1}^{n} X_i \to 1$ almost surely the limit is necessarily $0$.