Give an example of a quadruple (W, X, Y, Z) of random variables such that all of

ID: 3170529 • Letter: G

Question

Give an example of a quadruple (W, X, Y, Z) of random variables such that all of the following are true simultaneously: W and X have the same distribution, Z and Y have the same distribution, W + Z and X + Y do not have the same distribution

NOTE: (The point here is that even if you know the distribution of X and Y perfectly, you don’t know the distribution of X + Y unless you assume some sort of independence. This is why we had to jump through all the hoops Tuesday to show S 2 and X were independent before we could get the distribution of S 2 ).

Explanation / Answer

let W and X both follows chi square distribution with df p

Z and Y both follows chi square distribution q

moreover assume that W and Z are independent but X and Y are not

then by additive property W+Z follows a chi square distribution with df p+q

but since X and Y are not independent hence X+Y would not follow a chi square distribution with df p+q.

hence W and X have the same distribution, Z and Y have the same distribution, W + Z and X + Y do not have the same distribution

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Give an example of a quadruple (W, X, Y, Z) of random variables such that all of

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