Find the distribution of the sum of 25 independent Poisson random variables with the identical parameter λ = 5
Find the distribution of the sum of 25 independent Poisson random variables with the identical parameter λ = 5
Let $\displaystyle X_1,\dots,X_{25}$ be iid Poisson rv with parameter 5.
Then their mgf is : $\displaystyle M(t)=e^{5(e^t-1)}$
The mgf of $\displaystyle X=X_1+\dots+X_{25}$ is :
$\displaystyle M_X(t)=M_{X_1+\dots+X_{25}}(t)=M_{X_1}(t)\cdots M_{X_{25}}(t)=M(t) \cdots M(t)=[M(t)]^{25}$
$\displaystyle =\left(e^{5(e^t-1)}\right)^{25}=e^{25 \cdot 5(e^t-1)}=e^{125(e^t-1)}$
And this is the mgf of a Poisson distribution with parameter 125.