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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.

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Lectures prepared by: Elchanan Mossel Yelena Shvets

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  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

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