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Thread: [SOLVED] Y binomial r.v, show that the distribution of n-y is binomial r.v

  1. #1
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    [SOLVED] Y binomial r.v, show that the distribution of n-y is binomial r.v

    Let $\displaystyle Y$ be a binomial random variable with n trials and probability of success given by$\displaystyle p$. Show that $\displaystyle n-Y$ is a binomial random variable with $\displaystyle n$ trials and probability of success given by $\displaystyle 1-p$.

    Not sure how to take this, the mgf of Y is $\displaystyle m_y(t)=\left[p\cdot e^t + (1-p)\right] ^n$

    So the moment generating fucntion of $\displaystyle n-Y$ is
    $\displaystyle m_{n-Y}(t)=E(e^{nt-yt}$
    so im not sure how to expand this out. I get to
    $\displaystyle m_{n-Y}(t)=e^{nt}\cdot m_y(-t)$?
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  2. #2
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    doh!
    of course...

    $\displaystyle m_{n-Y}(t)=e^{nt} \cdot \left[p\cdot e^{-t} + (1-p)\right] ^n$
    $\displaystyle =\left[e^t \cdot p \cdot e^{-t}+(1-p)e^t\right]^n$
    $\displaystyle =\left[p+(1-p)e^t\right]^n
    $
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