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

1. ## [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
$m_{n-Y}(t)=e^{nt}\cdot m_y(-t)$?

2. 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
$