Jointly distributed random variables

**anitaK** · November 6th 2008, 09:37 PM

Hi can anyone help me with this q urgently
Suppose that Y1 and Y2 are independent Poisson variables with means m1 and m2 respectively.
Let X = Y1 + Y2. Show, using moment generating functions of independent variables or
otherwise, that X has a Poisson distribution with mean m = m1+ m2.

**mr fantastic** · November 6th 2008, 11:06 PM

Originally Posted by anitaK

Hi can anyone help me with this q urgently
Suppose that Y1 and Y2 are independent Poisson variables with means m1 and m2 respectively.
Let X = Y1 + Y2. Show, using moment generating functions of independent variables or
otherwise, that X has a Poisson distribution with mean m = m1+ m2.

You should know:

1. $m_X(t) = m_{Y_1}(t) \cdot m_{Y_2}(t)$ since Y1 and Y2 are independent random variables.

2. $m_{Y_1}(t) = e^{m_1 (e^t - 1)}$ and $m_{Y_2}(t) = e^{m_2 (e^t - 1)}$ .

It is simple to see that $m_X(t) = e^{(m_1 + m_2) (e^t - 1)}$ .

And since the moment generating function uniquely determines a probability distribution, ........

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