Resolved: "Whether the statement is true or false."

kuCAu

Answered question

2021-02-06

Whether the statement is true or false.

Answer & Explanation

Corben Pittman

Skilled2021-02-07Added 83 answers

We have to determine the sum of two independent Poisson random variables has a Poisson distribution.
Let X and Y are two Poisson random variables with parameters $λ_{1}$ and $λ_{2}$ .
Hence, the moment-generating functions are:
$M_{X} (t) = e^{- λ_{1} + λ_{1} e^{t}}$
$M_{Y} (t) = e^{- λ_{2} + λ_{2} e^{t}}$
By theorem,
$M_{X} + Y (t) = M x (t) M y (t)$
$M_{X} + Y (t) = e^{- λ_{1} + λ_{1} e^{t}} e^{- λ_{2} + λ_{2} e^{t}}$
$M_{X} + Y (t) = e^{(- λ_{1} + λ_{2})} e^{(- λ_{2} + λ_{2}) e^{t})}$
Hence, the moment-generating function of X+Y is a moment:
generating function of Poisson distribution with parameter $λ 1 + λ 2$ .
That means X+Yis alsofollows a Poisson distribution with parameter $λ 1 + λ 2$ .
Therefore, the statement sum of two independent Poisson random variables has a Poisson distribution is true.

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