Joint Distribution Functions/ part7

Conditional probability may be treated as a special case of conditional expectation. Namely, P ( A | X ) = E ( Y | X ) if Y is the indicator of A. Therefore the conditional probability also depends on the partition ?X generated by X rather than on X itself; P ( A | g(X) ) = P ( A | X ) = P ( A | ? ), ? = ?X = ?g(X).

On the other hand, conditioning on an event B is well-defined, provided that P ( B ) ? 0, irrespective of any partition that may contain B as one of several parts.

Conditional probability 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.

Probability density function and Conditional probability distribution
Example. A point of the sphere x2 + y2 + z2 = 1 is chosen at random according to the uniform distribution on the sphere [1] [2]. The random variables X, Y, Z are the coordinates of the random point. The joint density of X, Y, Z does not exist (since the sphere is of zero volume), but the joint density fX,Y of X, Y exists,