How to get the value of 'scaled' binomial distribution?
Andrew Henderson 
People kindly told me that there is not a equivalent popular distribution for $aX$ when $X$ is distributed as Binomial, but it is just a 'scaled' distribution. Here, $a$ is a positive constant.
Question
How to get the value of 'scaled' binomial distribution? Do I need to use $\Gamma$ function? A simpler way is appreciated.
To be more precise, I want to calculate $$g(x) = \frac{1}{|a|}f(\frac xa) = {n \choose x/a} p^{\frac xa} (1-p)^{(n-\frac xa)},$$ where $f(x)$ is $B(n,p)$ and $a$ is positive constant.
What is the simplest way to calculate $g(x)$?
Background
I'm considering this question under the same condition with this question, where $X$ is distributed as $Binomial$ and I want to know the distribution of $aX$. $a$ is a real and positive number.
I know $aX \sim \frac{1}{|a|}f(\frac xa)$ ($f$ means the distribution of $X$),so I tried to calculate the probability of $aX$ by $\frac{1}{|a|}f(\frac xa) = {n \choose x/a} p^{\frac ax} (1-p)^{(n-\frac xa)}$. But, this was not easy for me because $x/a$ is not a integer.
I think with $\Gamma$ function $nCr$ can be calculated for any real number $r (< n)$. But is there any straightforward way to calculate 'scaled' binomial distribution?
$\endgroup$ 51 Answer
$\begingroup$For a random variable $X$ of binomial distribution of parameters $n,p$ the possible values taken are $0,1,2, \cdots, n$ with probabilities$$P(X=k)={n \choose k}p^k(1-p)^{n-k}.$$ If $a$ is a real constant and $Y=aX$ then the valueas taken by $Y$ are $1a,2a,\cdots , na$ with probabilities $$P(Y=ak)=P(X=k).$$ I don't know if this answers the question.
EDITED
Perhaps, the reason of confusion is that there is no pdf. The cdf is $$F_Y(x)=P(aX<x)=P\left(X<\frac{x}{a}\right)=F_X\left(\frac{x}{a}\right).$$
If the derivative existed then one could get a formula similar to what you quoted for the Gaussian distribution.
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