Lectures prepared by: Elchanan Mossel Yelena Shvets Introduction to probability Stat 134 FAll 2005 Berkeley 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 - 1 a f(x)dx P(X  a,Y · b) = s - 1 a s - 1 b f(x) dx dy

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

x x Probability of Subsets Continuous Discrete

Constraints Continuous Discrete Non-negative: Integrates to 1:

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

Expectations ContinuousDiscrete Expectation of a function g(X): Covariance:

Expectations ContinuousDiscrete Expectation of a function g(X): Covariance:

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1}, Find the joint density f(x,y) such that P(X 2 dx, Y 2 dy) = f(x,y)dx dy. Find the marginals. Are X,Y independent? Compute: E(X),E(Y), P(Y<X); X’ » X, Y’ » Y & independent, find P(Y’<X’)? Questions: y=x 2 y=1 0

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1} Find the joint density f(x,y) such that P(X 2 dx, Y 2 dy) = f(x,y)dx dy. Solution: Since the density is uniform f(x,y) = c =1/area(D). D f(x,y) = ¾ for (x,y) 2 D; f(x,y) = 0 otherwise y=x 2 y=1 0

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1}, f(x,y) = ¾. y=x Find the marginals. y=1

Constraints Continuous Discrete Marginals: \

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1}, f(x,y) = ¾. y=x Solution: Find the marginals. y=1

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1} y=x y=1 Are X,Y independent? 0

Constraints Continuous Discrete Independence: for all x and y.

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1} y=x Solution: y=1 Are X,Y independent? X,Y are dependent! 0

Joint Distributions (X,Y) » Uniform{-1<X<1, X 2 · Y · 1} y=x Solution: y=x 0 Compute: E(X),E(Y), P(Y<X); A D-A

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

Joint Distributions X = Exp(1), Y = Exp(2), independent Find the joint density f(x,y) such that P(X 2 dx, Y 2 dy) = f(x,y)dx dy. Compute: P(X<2Y); Questions: f Y 0 X

Joint Distributions X = Exp(1), Y = Exp(2), independent Find the joint density. Questions: f Y 0 X Since X and Y are independent, we multiply the densities for X and Y For x ≥ 0, y ≥ 0

Joint Distributions X = Exp(1), Y = Exp(2), independent Questions: f Y 0 X We need to 1: Find the region x>2y Compute: P(X>2Y) X Y y = x/2

Joint Distributions X = Exp(1), Y = Exp(2), independent Questions: f Y 0 X We need to 2: Integrate over the region Compute: P(X>2Y) X Y X = 2Y