Find the distribution of the sum of 25 independent Poisson random variables with the identical parameterλ = 5

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- Apr 30th 2009, 09:55 AMplm2efind the distribution of the sum of 25 ind poisson rv's
Find the distribution of the sum of 25 independent Poisson random variables with the identical parameter

**λ = 5** - Apr 30th 2009, 09:59 AMMoo
- Apr 30th 2009, 10:06 AMplm2e
ok so, Mx(t) = e^[

**λ((e^t)- 1))]**

**i get e^[5((e^25) - 1)] = my ti 83 says overflow**

**i think i may be confused on what t is** - Apr 30th 2009, 10:08 AMplm2e
missed the product part on my last post, but im still concerned about what the t means in the mgf

- Apr 30th 2009, 10:26 AMMoo
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.