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Some additional Topics

Some additional Topics. Distributions of functions of Random Variables. Gamma distribution, c 2 distribution, Exponential distribution. Therorem.

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Some additional Topics

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  1. Some additional Topics

  2. Distributions of functions of Random Variables Gamma distribution, c2 distribution, Exponential distribution

  3. Therorem Let X and Y denote a independent random variables each having a gamma distribution with parameters (l,a1) and (l,a2). Then W = X + Y has a gamma distribution with parameters (l, a1 +a2). Proof:

  4. Recognizing that this is the moment generating function of the gamma distribution with parameters (l, a1 + a2) we conclude that W = X + Y has a gamma distribution with parameters (l, a1 + a2).

  5. Therorem(extension to n RV’s) Let x1, x2, … ,xndenote n independent random variables each having a gamma distribution with parameters (l,ai), i = 1, 2, …, n. Then W = x1 + x2 + … +xnhas a gamma distribution with parameters (l, a1 +a2 +… + an). Proof:

  6. Therefore Recognizing that this is the moment generating function of the gamma distribution with parameters (l, a1 + a2 +…+ an) we conclude that W = x1+ x2 + … + xnhas a gamma distribution with parameters (l, a1 + a2 +…+ an).

  7. Therorem Suppose that x is a random variable having a gamma distribution with parameters (l,a). Then W = axhas a gamma distribution with parameters (l/a, a). Proof:

  8. Special Cases • Let X and Y be independent random variables having an exponential distribution with parameter lthen X + Y has a gamma distribution with a = 2 andl • Let x1, x2,…, xn,be independent random variables having a exponential distribution with parameter lthen S = x1+ x2 +…+ xnhas a gamma distribution with a = n and l • Let x1, x2,…, xn,be independent random variables having a exponential distribution with parameter lthen • has a gamma distribution with a = n and nl

  9. Distribution of population – Exponential distribution Another illustration of the central limit theorem

  10. Special Cases -continued • Let X and Y be independent random variables having a c2 distribution with n1 and n2 degrees of freedom respectively then X + Y has a c2 distribution with degrees of freedom n1 + n2. • Let x1, x2,…, xn,be independent random variables having a c2 distribution with n1 ,n2 ,…, nndegrees of freedom respectively then x1+ x2 +…+ xnhas a c2 distribution with degrees of freedom n1 +…+ nn. Both of these properties follow from the fact that a c2 random variable with ndegrees of freedom is a Grandom variable with l = ½ and a = n/2.

  11. Recall If z has a Standard Normal distribution then z2 has a c2 distribution with 1 degree of freedom. Thus if z1, z2,…, znare independent random variables each having Standard Normal distribution then has a c2 distribution with ndegrees of freedom.

  12. Therorem Suppose that U1 and U2 are independent random variables and that U = U1 + U2 Suppose that U1 and U have a c2distribution with degrees of freedom n1andnrespectively. (n1 < n) Then U2 has a c2distribution with degrees of freedom n2 =n -n1 Proof:

  13. Q.E.D.

  14. Bivariate DistributionsDiscrete Random Variables

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

  16. Marginal distributions Conditional distributions

  17. The product rule for discrete distributions Independence

  18. Bayes rule for discrete distributions Proof:

  19. Continuous Random Variables

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

  21. Marginal distributions Conditional distributions

  22. The product rule for continuous distributions Independence

  23. Bayes rule for continuous distributions Proof:

  24. Example • Suppose that to perform a task we first have to recognize the task, then perform the task. • Suppose that the time to recognize the task, X, has an exponential distribution with l = ¼ (i,e, meanm= 1/l= 4 ) • Once the task is recognized the time to perform the task, Y, is uniform from X/2 to 2X. • Find the joint density of X and Y. • Find the conditional density of X given Y = y.

  25. Now and Thus

  26. Graph of non-zero region of f(x,y)

  27. Bayes rule for continuous distributions

  28. Conditional Expectation Let U = g(X,Y) denote any function of X and Y. Then is called the conditional expectation of U = g(X,Y) given X = x.

  29. Conditional Expectation and Variance More specifically is called the conditional expectation of Y given X = x. is called the conditional variance of Y given X = x.

  30. An Important Rule and where EXand VarX denote mean and variance with respect to the marginal distribution of X, fX(x).

  31. Proof Let U = g(X,Y) denote any function of X and Y. Then

  32. Now

  33. Example • Suppose that to perform a task we first have to recognize the task, then perform the task. • Suppose that the time to recognize the task, X, has an exponential distribution with l= ¼ (i,e, meanm= 1/l= 4 ) • Once the task is recognized the time to perform the task, Y, is uniform from X/2 to 2X. • Find E[XY]. • Find Var[XY].

  34. Solution

  35. Conditional Expectation: k (>2) random variables

  36. Definition then the conditional joint probability function of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk= xkis Let X1, X2, …, Xq, Xq+1…, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xq, xq+1 …, xk )

  37. Definition then the ConditionalExpectation of U given Xq+1 = xq+1 , …, Xk= xkis Let U = h( X1, X2, …, Xq, Xq+1…, Xk) Note this will be a function of xq+1 , …, xk.

  38. Example Determine the conditional expectation of U = X 2+ Y + Z given X = x, Y = y. Let X, Y, Z denote 3jointly distributed random variable with joint density function

  39. The marginal distribution of X,Y. Thus the conditional distribution of Z given X = x,Y = y is

  40. The conditional expectation of U = X 2+ Y + Z given X = x, Y = y.

  41. Thus the conditional expectation of U = X 2+ Y + Z given X = x, Y = y.

  42. The rule for Conditional Expectation Let (x1, x2, … , xq, y1, y2, … , ym) = (x, y) denote q + m random variables. Then

  43. Proof (in the simple case of 2 variables X and Y)

  44. hence

  45. Now

  46. The probability of a Gamblers ruin

  47. Suppose a gambler is playing a game for which he wins 1$ with probability p and loses 1$ with probability q. • Note the game is fair if p = q = ½. • Suppose also that he starts with an initial fortune of i$ and plays the game until he reaches a fortune of n$ or he loses all his money (his fortune reaches 0$) • What is the probability that he achieves his goal? What is the probability the he loses his fortune?

  48. Let Pi= the probability that he achieves his goal? Let Qi= 1 - Pi= the probability the he loses his fortune? Let X = the amount that he was won after finishing the game If the game is fair Then E [X] = (n – i )Pi + (– i )Qi = (n – i )Pi + (– i ) (1–Pi) = 0 or (n – i )Pi = i(1–Pi) and (n – i + i )Pi = i

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