18.44 Lecture Notes - Lecture 25: Random Variable, Probability Mass Function, Weighted Arithmetic Mean
Document Summary
It all starts with the de nition of conditional probability: p(a|b) = p(ab)/p(b). If x and y are jointly discrete random variables, we can use this to de ne a probability mass function for x given y = y. That is, we write p (x|y) = p{x = x|y = y} = In words: rst restrict sample space to pairs (x,y) with given y value. Then divide the original mass function by p (y) to obtain a probability mass function on the restricted space. We do something similar when x and y are continuous random variables. In that case we write f (x|y) = Often useful to think of sampling (x,y) as a two-stage process. First sample y from its marginal distribution, obtain y = y for some particular y. Then sample x from its probability distribution given y = y. Marginal law of x is weighted average of conditional laws.
