6 additional examples using joint distributions 143

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Unformatted text preview: t S has the binomial distribution with parameters m + n and p. Example 4.5.2 Suppose X and Y are independent random variables such that X has the Poisson distribution with parameter λ1 and Y has the Poisson distribution with parameter λ2 . Describe the distribution of S = X + Y. Solution: This problem can also be solved with a little thought and no calculation, as follows. Recall that the Poisson distribution is a limiting form of the binomial distribution with large n and small p. So let p be a very small positive number, and let m = λ1 /p and n = λ2 /p. (Round to the nearest integer if necessary so that m and n are integers.) Then the distribution of X is well approximated by the binomial distribution with parameters m and p, and the distribution of Y is 4.5. DISTRIBUTION OF SUMS OF RANDOM VARIABLES 137 well approximated by the binomial distribution with parameters n and p. So by Example 4.5.1, the distribution of X + Y is well approximated by the binomial distribution with parameters m + n and p. But m + n is large and p is...
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