Lectures prepared by:Elchanan MosselYelena Shvets

Introduction to probability

Stat 134 FAll 2005

Berkeley

Follows Jim Pitman’s book:

ProbabilitySection 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

.0

.1

.2

.3

.4

.5

-4 -3 -2 -1 0 1 2 3 4

2(x)-

21(x) e

2

2 2(x 4y )-

21(x,y) e

Probabilities single variable bivariate norm

.0

.1

.2

.3

.4

.5

-4 -3 -2 -1 0 1 2 3 4

P(X a) = s-1a f(x)dx P(X a,Y· b) =s-1

as-1bf(x) dx dy

x x-3

-2-1

01

23

0.

0.1

0.2

0.3

P(X=x, Y=y)=P(x,y)

Infinitesimal & Point Probability

Continuous Discrete

x x

-3-2

-10

12

3

0.

0.1

0.2

0.3

Probability of Subsets

Continuous Discrete

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} , 1

10

•Find the joint density f(x,y) such that P(X2dx, Y2dy) = 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=x2

y=1

0-1

Joint Distributions

(X,Y) » Uniform{-1<X<1, X2·Y· 1}.1

1 0

•Find the joint density f(x,y) such that P(X2dx, Y2dy) = 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=x2

y=1

0-1

Joint Distributions

(X,Y) » Uniform{-1<X<1, X2·Y· 1},

f(x,y) = ¾. y=x2

1

1 0

0

•Find the marginals.

y=1

-1

Constraints

Continuous Discrete•Marginals:

\

Joint Distributions

(X,Y) » Uniform{-1<X<1, X2·Y· 1},

f(x,y) = ¾. y=x2

1

1 0

0

Solution:

•Find the marginals.

y=1

-1

Joint Distributions

(X,Y) » Uniform{-1<X<1, X2·Y· 1}

y=x2

1

1 0

y=1

•Are X,Y independent?

0-1

Constraints

Continuous Discrete•Independence:

for all x and y.

Joint Distributions

(X,Y) » Uniform{-1<X<1, X2·Y· 1}

y=x2

1

1 0

Solution:

y=1

•Are X,Y independent?

•X,Y are dependent! 0-1

Joint Distributions(X,Y) » Uniform{-1<X<1, X2·Y· 1}

y=x21

1 0

Solution:

y=x

0-1

•Compute: E(X),E(Y), P(Y<X); A

D-A

Joint Distributions

1

1 0

Solution:

y=x

0-1

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(X2dx, Y2dy) = f(x,y)dx

dy.

•Compute: P(X<2Y);

Questions:f

Y0

X

Joint Distributions

X = Exp(1), Y = Exp(2), independent

•Find the joint density.

Questions:f

Y0

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

Y0

XWe need to

1: Find the region x>2y

•Compute: P(X>2Y)

X

Yy = x/2

Joint Distributions

X = Exp(1), Y = Exp(2), independent

Questions:

f

Y0

XWe need to

2: Integrate over the region

•Compute: P(X>2Y)

X

Y X = 2Y