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

**Robb** · November 12th 2009, 12:26 AM

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
$m_{n-Y}(t)=e^{nt}\cdot m_y(-t)$ ?

**Robb** · November 12th 2009, 12:38 AM

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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