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Thread: Variance of a branching process

  1. #1
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    Variance of a branching process

    Let $\displaystyle N_j$ be the size of the jth generation of a branching process with $\displaystyle N_0 = 1$. Suppose the number of offspring produced by an individual has mean $\displaystyle \mu$ and variance $\displaystyle {\sigma}^2$. Let $\displaystyle \alpha_j = \frac{Var(N_j)}{{{\sigma}^2}{\mu}^{j-1}}$
    Show that $\displaystyle \alpha_j = \alpha_{j-1}+\mu^{j-1}$

    I get to:

    $\displaystyle \frac{Var(N_{j-1})}{\sigma^2\mu^{j-2}}+\mu^{j-1}$
    $\displaystyle =\frac{Var(N_{j-1})+{\sigma^2\mu^{2j-3}}}{{\sigma^2\mu^{j-2}}}$

    However as I am not seeing the expression for $\displaystyle Var(N_j)$ which enables you to solve this question
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    Re: Variance of a branching process

    Quote Originally Posted by FGT12 View Post
    Let $\displaystyle N_j$ be the size of the jth generation of a branching process with $\displaystyle N_0 = 1$. Suppose the number of offspring produced by an individual has mean $\displaystyle \mu$ and variance $\displaystyle {\sigma}^2$. Let $\displaystyle \alpha_j = \frac{Var(N_j)}{{{\sigma}^2}{\mu}^{j-1}}$
    Show that $\displaystyle \alpha_j = \alpha_{j-1}+\mu^{j-1}$

    I get to:

    $\displaystyle \frac{Var(N_{j-1})}{\sigma^2\mu^{j-2}}+\mu^{j-1}$
    $\displaystyle =\frac{Var(N_{j-1})+{\sigma^2\mu^{2j-3}}}{{\sigma^2\mu^{j-2}}}$

    However as I am not seeing the expression for $\displaystyle Var(N_j)$ which enables you to solve this question
    It would have been better if you had shown how you got to this result...

    But if I understand correctly what you just did, it's not the correct way. You have to start from $\displaystyle Var[N_j]=Var\left[\sum_{i=1}^{N_{j-1}} X_i\right]$, where $\displaystyle X_i$ is the number of offspring for the individual i, and the $\displaystyle N_i$ are iid with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$

    To do this, use the law of total variance, by conditioning to $\displaystyle N_{j-1}$ :

    $\displaystyle \begin{aligned}Var[N_j] &= E\left[Var\left[\sum_{i=1}^{N_{j-1}} X_i \bigg|N_{j-1}\right]\right]+Var\left[E\left[\sum_{i=1}^{N_{j-1}} X_i \bigg|N_{j-1}\right]\right] \\ &= E\left[\sum_{i=1}^{N_{j-1}} Var[X_i]\right]+Var\left[\sum_{i=1}^{N_{j-1}} E[X_i]\right] \\ &= E[N_{j-1}\cdot \sigma^2]+Var[N_{j-1}\cdot \mu] \\ &= \sigma^2 E[N_{j-1}]+\mu^2 Var[N_{j-1}]\end{aligned}$

    But it is commonly known that $\displaystyle E[N_{j-1}]=\mu^{j-1} N_0=\mu^{j-1}$ for a GW process (shown by induction).

    Thus $\displaystyle Var[N_j]=\sigma^2 \mu^{j-1}+\mu^2 Var[N_{j-1}]$, I can't find the mistake, so if you do....

    Hmm so I find $\displaystyle \alpha_j=1+\mu \alpha_{j-1}$
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