(a) Explain the meaning of a Poisson process, and use your explanation to argue what the...

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(a) Explain the meaning of a Poisson process, and use your explanation to argue what the distribution of the sum of two independent Poisson random variables must be. You do not need to prove anything explicitly. (b) Suppose that X and Y are independent Poisson random variables with param- eters A and u respectively. Prove that the conditional distribution of X, given that X+Y = n, is binomial. What are the parameters of this binomial distri- bution? Explain the result you obtain using the interpretation of the Poisson process and the binomial distribution. (c) Using the central limit theorem, in relation to appropriate Poisson random variables, prove that n² 1 as n→ o0. n" 1+ 1! 2! n! (a) Explain the meaning of a Poisson process, and use your explanation to argue what the distribution of the sum of two independent Poisson random variables must be. You do not need to prove anything explicitly. (b) Suppose that X and Y are independent Poisson random variables with param- eters A and u respectively. Prove that the conditional distribution of X, given that X+Y = n, is binomial. What are the parameters of this binomial distri- bution? Explain the result you obtain using the interpretation of the Poisson process and the binomial distribution. (c) Using the central limit theorem, in relation to appropriate Poisson random variables, prove that n² 1 as n→ o0. n" 1+ 1! 2! n!