Question

3. Use the probability generating function Px)(s) to find (a) E[X(10)] (b) VarX(10)] (c) P(X(5)-2) . ( 4.2 Probability Generating Functions The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking values 0,1, 2, Its particular strength is that it gives us an easy way of characterizing the distribution of X +Y when X and Y are independent In general it is difficult to find the distribution of a sum using the traditional probability function. The PGF transforms a sum into a product and enables it to be handled much more easily Sums of random variables are particarly important in the study of stochastic processes, because many stochastic processes are formed from the sum of a sequence of repeating steps: for example, the Gambler's Ruin from Section 2.7 The name probability generating function also gives us another clue to the role of the PGF. The PGF can be used to generate the probabilities of the distribution. This is generally tedious and is not often an efficient way of calculating probabilities. However, the fact that it can be done demonstrates that the PGF tells us everything there is to know about the distribution Definition: Let X be a discrete random variable taking values in the non-negative integers (0, 1,2,...J. The probability generating function (PGF) of X is Gx(s) E(s), for all s e R for which the sum converges Calculating the probability generating function Gx(s)-E()(x )

Answer 1

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