The determination of the probability distribution of

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Unformatted text preview: he basic variables is a difficult task. So, the evaluation of its moment generating functions can provide some useful information on the target variable. 5.1. Definition of Moment Generating Function The moment generating function of a random variable generating function (mgf) exists, its th is defined as derivative at the origin is the . If the moment th order central moment of . If X is a random variable taking integer values, then by definition, its moment generating function is: Similarly if is a random variable taking continuous values, the mgf is: Notes The basic concept is that if two random variables have identical moment generating functions, then they possess the same probability distribution. The procedure is to find the moment generating function and then compare it to all known ones to see if there is a match. This is most commonly done to see if a distribution approaches the normal distribution as the sample size tends to infinity. 5.2. Theorem 1 Let FX x and FY y are two cumulative distribution functions whose moments exist. Then, i. If and have bounded support, then FX u and FY u are equal for all for all integers if and only if … ii. If the moment generating functions exist and for all t in some neighborhood of , then FX u FY u for all . 5.4 Theorem 2 Differentiating the equation of mgf for times, we obtain: 5.3. Corollary , then If If are independent and , then Problem 4. Let and , Find , given Solution. Here we get, that means So, has a normal distribution with mean and variance . Proble...
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