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Jointly distributed Random variables

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  1. Jointly distributed Random variables Multivariate distributions

  2. Quite often there will be 2 or more random variables (X, Y, etc) defined for the same random experiment. Example: A bridge hand (13 cards) is selected from a deck of 52 cards. X = the number of spades in the hand. Y = the number of hearts in the hand. In this example we will define: p(x,y) = P[X = x, Y = y]

  3. The function p(x,y) = P[X = x, Y = y] is called the joint probability function of X and Y.

  4. Note: The possible values of X are 0, 1, 2, …, 13 The possible values of Y are also 0, 1, 2, …, 13 and X + Y ≤ 13. The number of ways of choosing the yhearts for the hand The number of ways of choosing the xspades for the hand The number of ways of completing the hand with diamonds and clubs. The total number of ways of choosing the 13 cards for the hand

  5. Table: p(x,y)

  6. Bar graph: p(x,y) p(x,y) y x

  7. Example: A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. Now p(x,y) = P[X = x, Y = y] The possible values of X are 0, 1, 2, 3, 4, 5. The possible values of Y are 0, 1, 2, 3, 4, 5. and X + Y ≤ 5

  8. A typical outcome of rolling a die n = 5 times will be a sequence F5FF6 where F denotes the outcome {1,2,3,4}. The probability of any such sequence will be: where x = the number of sixes in the sequence and y = the number of fives in the sequence

  9. Now p(x,y) = P[X = x, Y = y] Where K = the number of sequences of length 5 containing x sixes and y fives.

  10. Thus p(x,y) = P[X = x, Y = y] if x + y ≤ 5.

  11. Table: p(x,y)

  12. Bar graph: p(x,y) p(x,y) y x

  13. General properties of the joint probability function; p(x,y) = P[X = x, Y = y]

  14. Example: A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. What is the probability that we roll more sixes than fives i.e. what is P[X > Y]?

  15. Table: p(x,y)

  16. Marginal and conditional distributions

  17. Definition: Let X and Y denote two discrete random variables with joint probability function p(x,y) = P[X = x, Y = y] Then pX(x) = P[X = x] is called the marginal probability function of X. and pY(y) = P[Y = y] is called the marginal probability function of Y.

  18. Note: Let y1, y2, y3, … denote the possible values of Y. Thus the marginal probability function of X, pX(x) is obtained from the joint probability function of X and Y by summing p(x,y) over the possible values of Y.

  19. Also

  20. Example: A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears.

  21. Conditional Distributions

  22. Definition: Let X and Y denote two discrete random variables with joint probability function p(x,y) = P[X = x, Y = y] Then pX |Y(x|y) = P[X = x|Y = y] is called the conditional probability function of X given Y = y and pY |X(y|x) = P[Y = y|X = x] is called the conditional probability function of Y given X = x

  23. Note and

  24. Marginal distributions describe how one variable behaves ignoring the other variable. • Conditional distributions describe how one variable behaves when the other variable is held fixed

  25. Example: A die is rolled n = 5 times X = the number of times a “six”appears. Y = the number of times a “five”appears. y x

  26. The conditional distribution of X given Y = y. pX |Y(x|y) = P[X = x|Y = y] y x

  27. The conditional distribution of Y given X = x. pY |X(y|x) = P[Y = y|X = x] y x

  28. Example A Bernoulli trial (S - p, F – q = 1 – p) is repeated until two successes have occurred. X = trial on which the first success occurs and Y = trial on which the 2nd success occurs. Find the joint probability function of X, Y. Find the marginal probability function of X and Y. Find the conditional probability functions of Y given X = x and X given Y = y,

  29. Solution x y A typical outcome would be: FFF…FSFFF…FS x - 1 y – x - 1

  30. p(x,y) - Table

  31. The marginal distribution of X This is the geometric distribution

  32. The marginal distribution of Y This is the negative binomial distribution with k = 2.

  33. The conditional distribution of X given Y = y This is the geometric distribution with time starting at x.

  34. The conditional distribution of Y given X = x This is the uniform distribution on the values 1, 2, …(y – 1)

  35. SummaryDiscrete Random Variables

  36. The joint probability function; p(x,y) = P[X = x, Y = y]

  37. Continuous Random Variables

  38. Definition: Two random variable are said to have joint probability density function f(x,y) if

  39. If then defines a surface over the x – y plane

  40. Multiple Integration

  41. A f(x,y) A

  42. y d c x a b If the region A = {(x,y)| a ≤ x ≤ b, c ≤ y ≤ d} is a rectangular region with sides parallel to the coordinate axes: Then A f(x,y)

  43. To evaluate A First evaluate the inner integral Then evaluate the outer integral f(x,y)

  44. y d c x a b dy y = area under surface above the line where y is constant Infinitesimal volume under surface above the line where y is constant f(x,y)

  45. The same quantity can be calculated by integrating first with respect to y, than x. A First evaluate the inner integral Then evaluate the outer integral f(x,y)

  46. y d c x a b dx x = area under surface above the line where x is constant Infinitesimal volume under surface above the line where x is constant f(x,y)

  47. Example: Compute Now f(x,y)

  48. The same quantity can be computed by reversing the order of integration f(x,y)

  49. Integration over non rectangular regions

  50. y Suppose the region A is defined as follows A = {(x,y)| a(y)≤ x ≤ b(y),c ≤ y ≤ d} d c x Then b(y) a(y) A