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Math Help - expected value and variance

  1. #1
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    expected value and variance

    A certain organism produces an average of \mu offsprings with variance \sigma ^2. Each offspring produces likewise. What would be the expected number of offspring and its variance in the third generation?
    If X_{i} = # of offspring in ith generation, I figured E(X_{3})= \mu ^2 and Var(X_{3}) = \mu ^2 \sigma ^2
    I'm pretty sure of expected, but not that sure of variance. I used the fact that if Y= a+bX then Var(Y)=b^{2}Var(X)
    Can somebody confirm this? Thanks.
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    Hello,

    This is similar to a Galton-Watson process (if it is assumed that the production of offsprings are independent).
    And assuming that the "parents" die once they give their offspring, we have X_{n+1}=\sum_{k=1}^{X_n} \xi_{k,n+1}, where (\xi_{k,n})_n is an iid sequence with mean \mu and variance \sigma^2

    And you're looking for E[X_3] and Var[X_3] (where X_0 - or X_1 ? - is 1)

    It's easy to compute E[X_{n+1}|X_n] (using basic properties of conditional expectations), and it gives the result you got by taking the expectation again.

    And for the variance, I suggest you use the law of total variance : Law of total variance - Wikipedia, the free encyclopedia

    If you have trouble, let us know
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