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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. using or

10. then

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

12. We will use

13. Note: Also

14. Note: Also

15. Equating coefficients of tk, we get

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

17. 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.

18. 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 .

19. 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

20. 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

21. 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

22. 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

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

24. 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

25. 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

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

27. Thus has a normal distribution with mean and variance

28. 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

29. Sampling distribution of Population

30. 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

31. 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

32. 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

33. 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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