18.44 Lecture Notes - Lecture 23: Conditional Probability, Beta Distribution

• School: MIT
• Department: Mathematics
• Course: 18.44
• Professor: Scott Sheffield

Document Summary

Suppose x and y are chosen uniformly on the semicircle {(x,y) : x + y 1 , x 0}. Answer: f (x) = 1 if x [0,1] (zero otherwise). Answer: f (x) = 2x if x [0,1] (zero otherwise). Both { = 0} and {y = 0} describe the same probability zero event. But our interpretation of what it means to condition on this event is different in these two cases. Conditioning on (x,y) belonging to a ( , ) wedge is very different from conditioning on (x , y ) belonging to a. Suppose i choose n random variables x , x , , x uniformly at random on [0, 1], independently of each other. The n-tuple (x , x , , x ) has a constant density function on the n-dimensional cube [0, 1] Let"s say x and y have joint probability density function f(x, y)