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Use of moment generating functions

Use of moment generating functions. Definition. Let X denote a random variable with probability density function f ( x ) if continuous (probability mass function p ( x ) if discrete) Then m X ( t ) = the moment generating function of X.

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Use of moment generating functions

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  1. Use of moment generating functions

  2. Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function p(x) if discrete) Then mX(t) = the moment generating function of X

  3. The distribution of a random variable X is described by either • The density function f(x) if X continuous (probability mass function p(x) if X discrete), or • The cumulative distribution function F(x), or • The moment generating function mX(t)

  4. Properties • mX(0) = 1

  5. Let X be a random variable with moment generating function mX(t). Let Y = bX + a Then mY(t) = mbX + a(t) = E(e [bX + a]t) = eatmX (bt) • Let X and Y be two independent random variables with moment generating function mX(t) and mY(t) . Then mX+Y(t) = mX (t) mY (t)

  6. Let X and Y be two random variables with moment generating function mX(t) and mY(t) and two distribution functions FX(x) and FY(y) respectively. Let mX (t) = mY (t) then FX(x) = FY(x). This ensures that the distribution of a random variable can be identified by its moment generating function

  7. M. G. F.’s - Continuous distributions

  8. M. G. F.’s - Discrete distributions

  9. Moment generating function of the gamma distribution where

  10. using or

  11. then

  12. Moment generating function of the Standard Normal distribution where thus

  13. We will use

  14. Note: Also

  15. Note: Also

  16. Equating coefficients of tk, we get

  17. Using of moment generating functions to find the distribution of functions of Random Variables

  18. Example Suppose that X has a normal distribution with mean mand standard deviation s. Find the distribution of Y = aX + b Solution: = the moment generating function of the normal distribution with mean am + b and variance a2s2.

  19. Thus Y = aX + b has a normal distribution with mean am + b and variance a2s2. Special Case: the z transformation Thus Z has a standard normal distribution .

  20. Example Suppose that X and Y are independent eachhaving a normal distribution with means mX and mY , standard deviations sX and sY Find the distribution of S = X + Y Solution: Now

  21. or = the moment generating function of the normal distribution with mean mX + mY and variance Thus Y = X + Y has a normal distribution with mean mX + mY and variance

  22. Example Suppose that X and Y are independent eachhaving a normal distribution with means mX and mY , standard deviations sX and sY Find the distribution of L = aX + bY Solution: Now

  23. or = the moment generating function of the normal distribution with mean amX + bmY and variance Thus Y = aX + bY has a normal distribution with mean amX + BmY and variance

  24. a = +1 and b = -1. Special Case: Thus Y = X - Y has a normal distribution with mean mX - mY and variance

  25. Example (Extension to n independent RV’s) Suppose that X1, X2, …, Xn are independent eachhaving a normal distribution with means mi, standard deviations si (for i = 1, 2, … , n) Find the distribution of L = a1X1 + a1X2 + …+ anXn Solution: (for i = 1, 2, … , n) Now

  26. or = the moment generating function of the normal distribution with mean and variance Thus Y = a1X1 + … + anXnhas a normal distribution with mean a1m1+ …+ anmn and variance

  27. Special case: In this case X1, X2, …, Xn is a sample from a normal distribution with mean m, and standard deviations s, and

  28. Thus has a normal distribution with mean and variance

  29. Summary If x1, x2, …, xn is a sample from a normal distribution with mean m, and standard deviations s, then has a normal distribution with mean and variance

  30. Sampling distribution of Population

  31. The Central Limit theorem If x1, x2, …, xn is a sample from a distribution with mean m, and standard deviations s, then if n is large has a normal distribution with mean and variance

  32. Proof: (use moment generating functions) We will use the following fact: Let m1(t), m2(t), … denote a sequence of moment generating functions corresponding to the sequence of distribution functions: F1(x) , F2(x), … Let m(t) be a moment generating function corresponding to the distribution function F(x) then if then

  33. Let x1, x2, … denote a sequence of independent random variables coming from a distribution with moment generating function m(t) and distribution function F(x). Let Sn = x1 + x2 + … + xn then

  34. Is the moment generating function of the standard normal distribution Thus the limiting distribution of z is the standard normal distribution Q.E.D.

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