1 / 25

Lectures prepared by: Elchanan Mossel Yelena Shvets

Lectures prepared by: Elchanan Mossel Yelena Shvets. 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.

erek
Download Presentation

Lectures prepared by: Elchanan Mossel Yelena Shvets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lectures prepared by:Elchanan MosselYelena Shvets

  2. Joint Desity The density functionf(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. Densitiessingle variable bivariate norm

  5. Probabilitiessingle 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 Discrete x x P(X=x, Y=y)=P(x,y)

  7. Probability of Subsets Continuous Discrete x x

  8. ConstraintsContinuous Discrete • Non-negative: • Integrates to 1:

  9. ConstraintsContinuous Discrete • Marginals: • \ • Independence: for all x and y.

  10. ExpectationsContinuousDiscrete • Expectation of a function g(X): • Covariance:

  11. ExpectationsContinuousDiscrete • Expectation of a function g(X): • Covariance:

  12. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} , y=1 1 Questions: y=x2 • Find the joint density f(x,y) such that P(X2dx, Y2dy) = f(x,y)dx dy. 0 • Find the marginals. -1 1 0 • Are X,Y independent? • Compute: E(X),E(Y), P(Y<X); • X’»X, Y’»Y & independent, find P(Y’<X’)?

  13. 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 Solution: Since the density is uniform f(x,y) = c =1/area(D). 0 -1 1 0 f(x,y) = ¾ for (x,y) 2 D; f(x,y) = 0 otherwise

  14. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1}, f(x,y) = ¾. y=1 1 y=x2 • Find the marginals. 0 -1 1 0

  15. ConstraintsContinuous Discrete • Marginals: • \

  16. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1}, f(x,y) = ¾. y=1 1 y=x2 • Find the marginals. Solution: 0 -1 1 0

  17. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} y=1 1 y=x2 • Are X,Y independent? 0 -1 1 0

  18. ConstraintsContinuous Discrete • Independence: for all x and y.

  19. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} y=1 1 • Are X,Y independent? y=x2 Solution: 0 • X,Y are dependent! -1 1 0

  20. Joint Distributions (X,Y) » Uniform{-1<X<1, X2·Y· 1} 1 y=x2 y=x A • Compute: E(X),E(Y), P(Y<X); Solution: 0 -1 1 0 D-A

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

  22. f Y 0 X Joint Distributions X = Exp(1), Y = Exp(2), independent Questions: • Find the joint density f(x,y) such that P(X2dx, Y2dy) = f(x,y)dx dy. • Compute: P(X<2Y);

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

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

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

More Related