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

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• Nov 12th 2009, 12:26 AM
Robb
[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
$\displaystyle m_{n-Y}(t)=e^{nt}\cdot m_y(-t)$?
• Nov 12th 2009, 12:38 AM
Robb
doh!
of course...

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