Let X and Y be independent random variables following Poisson distributions, each with parameter λ = 1. Show that the distribution of Z = X + Y is Poisson with parameter λ = 2. using convolution formula
Probability Mass Function of X Poisson with parameter = 1 is given by:
Probability Mass Function of Y Poisson with parameter = 1 is given by:
To find the distribution of Z = X + Y
By Convolution Theorem:
Multiplying and dividing by z!, we get:
By Binomial Theorem, we have:
Substituting (2), equation (1) becomes:
Thus, we prove that the distribution of Z = X + Y is Poisson with parameter = 2.
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