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

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).

3. Joint Desity

4. Densities single variable bivariate norm

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

6. Infinitesimal & Point Probability Continuous Discretex x
P(X=x, Y=y)=P(x)

7. Probability of Subsets Continuous Discretex x

8. Constraints Continuous Discrete
Non-negative:
Integrates to 1:

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

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

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

12. 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’)?

13. 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.

14. 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

15. Constraints Continuous Discrete
Marginals: \

16. 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

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

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

19. 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!

20. 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

21. 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.