Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley Lectures prepared by: Elchanan Mossel Yelena Shvets Follows Jim Pitman’s book: Probability Section 5.2

Joint Desity The density function f(x,y) for a pair of RVs X and Y is the density of probability per area of (X,Y) near (x,y).

Joint Desity

Densities single variable bivariate norm

Probabilities single variable bivariate norm P(X £ a) = s-1a f(x)dx P(X £ a,Y· b) =s-1as-1bf(x) dx dy

Infinitesimal & Point Probability Continuous Discrete x x P(X=x, Y=y)=P(x)

Probability of Subsets Continuous Discrete x x

Constraints Continuous Discrete Non-negative: Integrates to 1:

Constraints Continuous Discrete Marginals: \ Independence: for all x and y.

Expectations Continuous Discrete Expectation of a function g(X): Covariance:

Expectations Continuous Discrete Expectation of a function g(X): Covariance:

Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} , y=1 Questions: y=x2 Find the joint density f(x,y) such that P(X2dx, Y2dy) = f(x,y)dx dy. Find the marginals. -1 1 Are X,Y independent? Compute: E(X),E(Y), P(Y<X); X’»X, Y’»Y & independent, find P(Y’<X’)?

Find the joint density f(x,y) such that P(X2dx, Y2dy) = f(x,y)dx dy. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1}. y=1 1 Find the joint density f(x,y) such that P(X2dx, Y2dy) = f(x,y)dx dy. y=x2 D -1 1 Solution: Since the density is uniform f(x,y) = c =1/area(D). f(x,y) = ¾.

Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1}, y=1 f(x,y) = ¾. y=1 1 y=x2 Find the marginals. -1 1

Constraints Continuous Discrete Marginals: \

Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1}, y=1 f(x,y) = ¾. y=1 1 y=x2 Find the marginals. Solution: -1 1

Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} y=1 Are X,Y independent? y=x2 -1 1

Constraints Continuous Discrete \ Independence: for all x and y.

Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} y=1 Are X,Y independent? y=x2 Solution: -1 1 X,Y are dependent!

Compute: E(X),E(Y), P(Y<X); Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} 1 y=x2 y=x Compute: E(X),E(Y), P(Y<X); A Solution: -1 1 D-A

X’»X, Y’»Y & X’,Y’ are independent, find P(Y’<X’)? Joint Distributions X’»X, Y’»Y & X’,Y’ are independent, find P(Y’<X’)? 1 y=x Solution: A -1 1 We need to integrate this density over the indicated region A = the subset of the rectangle [-1,1]£[0,1] where y<x.