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Random sums of rvs

  1. Feb 10, 2012 #1
    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] ?
     
  2. jcsd
  3. Feb 10, 2012 #2

    Ray Vickson

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    Homework Helper

    Z is a continuous random variable, so does not have a discrete generating function M_Z(z). You should be looking at its MGF [itex] M_Y(u) = E e^{u Y}, [/itex] or its Laplace transform [itex] L_Y(s) = E e^{-s Y}. [/itex] You almost had it right, but you switched the roles of the two types of transforms.

    Another, perhaps more direct approach is to get the density [itex] f_Y(t)[/itex] of Y from
    [tex] f_Y(t) dt = \sum_{n=1}^{\infty} P\{N=n\} P\{ Y \in (t,t+dt) | N=n \}, [/tex]
    and noting that given {N=n}, Y has an n-Erlang distribution.

    RGV
     
  4. Feb 11, 2012 #3
    I'm still confused... this is what I was doing:
    [tex]M_Z(z)=E(e^{zZ})=E[E(e^{zZ}|N)]=E[(Ee^{zX_1})(Ee^{zX_2})\ldots(Ee^{zX_N})][/tex]
    [tex]=E[(Ee^{zX})^N]=E[(M_X(z))^N]=G_N(M_X(z))[/tex]
    Where am I going wrong? Should I be doing this:
    [tex]M_Z(s)=M_X(G_N(s))[/tex]
     
    Last edited: Feb 11, 2012
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