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Published byJesse Patterson Modified over 6 years ago

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Use of moment generating functions

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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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The distribution of a random variable X is described by either 1.The density function f(x) if X continuous (probability mass function p(x) if X discrete), or 2.The cumulative distribution function F(x), or 3.The moment generating function m X (t)

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Properties 1. m X (0) = 1 2. 3.

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4. Let X be a random variable with moment generating function m X (t). Let Y = bX + a Then m Y (t) = m bX + a (t) = E(e [bX + a]t ) = e at m X (bt) 5. Let X and Y be two independent random variables with moment generating function m X (t) and m Y (t). Then m X+Y (t) = m X (t) m Y (t)

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

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M. G. F.’s - Continuous distributions

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M. G. F.’s - Discrete distributions

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Moment generating function of the gamma distribution where

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using or

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then

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Moment generating function of the Standard Normal distribution where thus

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We will use

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Note: Also

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Note: Also

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Equating coefficients of t k, we get

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

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Example Suppose that X has a normal distribution with mean and standard deviation . Find the distribution of Y = aX + b Solution: = the moment generating function of the normal distribution with mean a + b and variance a 2 2.

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Thus Z has a standard normal distribution. Special Case: the z transformation Thus Y = aX + b has a normal distribution with mean a + b and variance a 2 2.

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Example Suppose that X and Y are independent each having a normal distribution with means X and Y, standard deviations X and Y Find the distribution of S = X + Y Solution: Now

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

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Example Suppose that X and Y are independent each having a normal distribution with means X and Y, standard deviations X and Y Find the distribution of L = aX + bY Solution: Now

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or = the moment generating function of the normal distribution with mean a X + b Y and variance Thus Y = aX + bY has a normal distribution with mean a X + B Y and variance

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

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Example (Extension to n independent RV’s) Suppose that X 1, X 2, …, X n are independent each having a normal distribution with means i, standard deviations i (for i = 1, 2, …, n) Find the distribution of L = a 1 X 1 + a 1 X 2 + …+ a n X n Solution: Now (for i = 1, 2, …, n)

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or = the moment generating function of the normal distribution with mean and variance Thus Y = a 1 X 1 + … + a n X n has a normal distribution with mean a 1 1 + …+ a n n and variance

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In this case X 1, X 2, …, X n is a sample from a normal distribution with mean , and standard deviations and Special case:

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Thus and variance has a normal distribution with mean

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If x 1, x 2, …, x n is a sample from a normal distribution with mean , and standard deviations then Summary and variance has a normal distribution with mean

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Population Sampling distribution of

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If x 1, x 2, …, x n is a sample from a distribution with mean , and standard deviations then if n is large The Central Limit theorem and variance has a normal distribution with mean

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We will use the following fact: Let m 1 (t), m 2 (t), … denote a sequence of moment generating functions corresponding to the sequence of distribution functions: F 1 (x), F 2 (x), … Let m(t) be a moment generating function corresponding to the distribution function F(x) then if Proof: (use moment generating functions) then

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Let x 1, x 2, … denote a sequence of independent random variables coming from a distribution with moment generating function m(t) and distribution function F(x). Let S n = x 1 + x 2 + … + x n then

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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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