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发帖数: 268 | 1 如果{Yn|X}是iid的，X～Bernoulli(1/2)，{Yn}是iid的吗？
反过来，如果{Yn}是iid的，X～Bernoulli(1/2)，{Yn|X}是iid的吗？ | m**********r
发帖数: 507 | 2 1. No. e.g., n=2 set Y1 = Y2 = X = Ber(1/2)
-> Y1|X is independent of Y2|X
(1) P(Y1=1, Y2=1|X=1) = 1 = P(Y1=1|X=1) P(Y2=1|X=1)
(2) P(Y1=1, Y2=0|X=1) = 0 = P(Y1=1|X=1) P(Y2=0|X=1)
(3) P(Y1=0, Y2=1|X=1) = 0 = P(Y1=0|X=1) P(Y2=1|X=1)
(4) P(Y1=0, Y2=0|X=1) = 0 = P(Y1=0|X=1) P(Y2=0|X=1)
similarly check ALL combinations of Y1 and Y2 conditioning on the event X=0
-> marginally, both Y1 and Y2 are Ber(1/2)
Thus P(Y1=1) * P(Y2=1) = (1/2) * (1/2) = 1/4
-> But Y1 is NOT independent of Y2, because
P(Y1=1, Y2=1) = P(Y1=1, Y2=1|X=1)P(X=1) = 1 * (1/2) = 1/2
which is NOT equal to P(Y1=1) * P(Y2=1) = 1/4
2. No. e.g., n=2.
-> Suppose Y1 and Y2 are two independent copies of Ber(1/2)
-> Set X to be:
X=1 if Y1=Y2;
X=0 otherwise
-> Check X ~ Ber(1/2)
P(X=1) = P(Y1=Y2) = P(Y1=Y2=1) + P(Y1=Y2=0) = 1/2
P(X=0) = 1-P(X=1) = 1/2
-> Check Y1|X is NOT independent of Y2|X
P(Y1=1, Y2=1|X=1) = P(Y1=1, Y2=1, X=1)/P(X=1)
= P(Y1=1, Y2=1)/P(X=1) = (1/4)/(1/2)=1/2
But by symmetry of Y1 and Y2, P(Y1=1|X=1)=P(Y2=1|X=1), so that
P(Y1=1|X=1)P(Y2=1|X=1) = [P(Y1=1|X=1)]^2
= [P(Y1=1, Y2=1|X=1)]^2 = (1/2)^2 = 1/4 !! |
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