suppose X~Bin(100,p), Y~Bin(80,p) and X, Y are independent. Let Z= X+Y. Show using the moment generating function method that Z~Bin(180,p)

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We need to prove that the sum of two independent binomial random variables follows a binomial distribution using the moment generating function method. Given X follows binomial distribution with parameters 100 and p, Y follows binomial distribution with parameters 80 and p, and they are independent. We want to show that Z equals X plus Y follows binomial distribution with parameters 180 and p. The moment generating function of a binomial distribution with parameters n and p is given by the formula shown.