Conditional Distribution
Given X = x, the conditional distribution of Y is
for 0 ≤ y ≤ min ( 3, x ). It is the hypergeometric distribution H ( x; 3, 7 ), or equivalently, H ( 3; x, 10-x ). The corresponding expectation 0.3 x, obtained from the general formula
for H ( n; R, W ), is nothing but the conditional expectation E (Y | X = x) = 0.3 x.
Treating H ( X; 3, 7 ) as a random distribution (a random vector in the four-dimensional space of all measures on {0,1,2,3}), one may take its expectation, getting the unconditional distribution of Y, — the binomial distribution Bin ( 3, 0.5 ). This fact amounts to the equality
for y = 0,1,2,3; just the law of total probability.
Read more about this topic: Conditioning (probability), Conditioning On The Discrete Level
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