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Math Help - [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 Y be a binomial random variable with n trials and probability of success given by  p. Show that n-Y is a binomial random variable with n trials and probability of success given by 1-p.

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

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

    m_{n-Y}(t)=e^{nt} \cdot \left[p\cdot e^{-t} + (1-p)\right] ^n
    =\left[e^t \cdot p \cdot e^{-t}+(1-p)e^t\right]^n
    =\left[p+(1-p)e^t\right]^n<br />
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