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Moment Generating Functions. The Uniform distribution from a to b. Continuous Distributions. The Normal distribution (mean m , standard deviation s ). The Exponential distribution. Weibull distribution with parameters a and b . The Weibull density, f ( x ). ( a = 0.9, b = 2).

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Presentation Transcript
the weibull density f x
The Weibull density, f(x)

(a= 0.9, b= 2)

(a= 0.7, b= 2)

(a= 0.5, b= 2)

the gamma distribution
The Gamma distribution

Let the continuous random variable X have density function:

Then X is said to have a Gamma distribution with parameters aand l.

slide9
X is discrete

X is continuous

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the kthcentralmoment of X

wherem = m1 = E(X) = the first moment of X .

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The Poisson distribution (parameter l)

The moment generating function of X , mX(t) is:

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The Exponential distribution (parameter l)

The moment generating function of X , mX(t) is:

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The Standard Normal distribution (m = 0, s = 1)

The moment generating function of X , mX(t) is:

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We will now use the fact that

We have completed the square

This is 1

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The Gamma distribution (parameters a, l)

The moment generating function of X , mX(t) is:

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We use the fact

Equal to 1

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mX(0) = 1

Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1

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The moments for the exponential distribution can be calculated in an alternative way. This is note by expanding mX(t) in powers of t and equating the coefficients of tk to the coefficients in:

Equating the coefficients of tk we get:

the moments for the standard normal distribution
The moments for the standard normal distribution

We use the expansion of eu.

We now equate the coefficients tk in:

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For even 2k:

If k is odd: mk= 0.

summary

Summary

Moments

Moment generating functions

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Moments of Random Variables

The moment generating function

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The Binomial distribution (parameters p, n)

Examples

  • The Poisson distribution (parameter l)
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The Exponential distribution (parameter l)

  • The Standard Normal distribution (m = 0, s = 1)
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The Gamma distribution (parameters a, l)

  • The Chi-square distribution (degrees of freedom n)

(a = n/2, l = 1/2)

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The log of Moment Generating Functions

Let lX (t) = ln mX(t) = the log of the moment generating function

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Thus lX (t) = ln mX(t) is very useful for calculating the mean and variance of a random variable