(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[itex]Z=X_1+\ldots+X_N[/itex], where:

[itex]X_i\sim_{iid}\,\text{Exponential}(\lambda)[/itex]

[itex]N\sim\,\text{Geometric}_1(p)[/itex]

For all [itex]i,\,N[/itex] and [itex]X_i[/itex] are independent.

Find the probability distribution of [itex]Z[/itex]

2. Relevant equations

[tex]G_N(t)=\frac{(1-p)t}{1-pt}[/tex]

[tex]M_X(t)=\frac{\lambda}{\lambda-t}[/tex]

3. The attempt at a solution

[tex]M_Z(z)=G_N(M_X(z))=\frac{(1-p)\left(\frac{\lambda}{\lambda-z}\right)}{1-p\left(\frac{ \lambda}{\lambda-z}\right)}[/tex]

[tex]Z\sim\,\text{Geometric}_1\left(p \frac{ \lambda}{\lambda-z}\right)[/tex]

Is that even correct? Should I be looking for [itex]E[Z][/itex] and [itex]V[Z][/itex] ?

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# Homework Help: Random sums of rvs

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