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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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## Moment Generating Functions

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**The Uniform distribution from a to b**Continuous Distributions**The Weibull density, f(x)**(a= 0.9, b= 2) (a= 0.7, b= 2) (a= 0.5, b= 2)**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.**X is discrete**X is continuous**the kthcentralmoment of X**wherem = m1 = E(X) = the first moment of X .**The Poisson distribution (parameter l)**The moment generating function of X , mX(t) is:**The Exponential distribution (parameter l)**The moment generating function of X , mX(t) is:**The Standard Normal distribution (m = 0, s = 1)**The moment generating function of X , mX(t) is:**We will now use the fact that**We have completed the square This is 1**The Gamma distribution (parameters a, l)**The moment generating function of X , mX(t) is:**We use the fact**Equal to 1**mX(0) = 1**Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1**Property 3 is very useful in determining the moments of a**random variable X. Examples**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**We use the expansion of eu. We now equate the coefficients tk in:**For even 2k:**If k is odd: mk= 0.**Summary**Moments Moment generating functions**Moments of Random Variables**The moment generating function**The Binomial distribution (parameters p, n)**Examples • The Poisson distribution (parameter l)**The Exponential distribution (parameter l)**• The Standard Normal distribution (m = 0, s = 1)**The Gamma distribution (parameters a, l)**• The Chi-square distribution (degrees of freedom n) (a = n/2, l = 1/2)**Properties of Moment Generating Functions**• mX(0) = 1**The log of Moment Generating Functions**Let lX (t) = ln mX(t) = the log of the moment generating function**Thus lX (t) = ln mX(t) is very useful for calculating the**mean and variance of a random variable

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