<s>[INST] Binomial distribution closed by linear operations? I understand that normal distributions are closed by linear operations, but is binomial distribution as well?
In other words, if $X$ is a binomial random variable, does $aX + b$ follow a binomial distribution for any real number $a$ and $b$?
 [/INST] Is Y=aX+b binomial for all real a,b? No, if a=2 then the support of Y is 0,2,4,...,2n which is not binomial.
Do the values of Y follow "binomial probabilities"? Yes. If $X\sim Bin(n,p), Y=aX+b$, then the support of Y is $aX+b, X=0,1,...,n$ and the pmf is $f(Y)={n\choose \frac{y-b}{a}}p^{\frac{y-b}{a}}(1-p)^{1-\frac{y-b}{a}}$.
Example
$X\sim Bin(3,.4)$, $Y=2X+1$
$X\begin{cases}0&.6^3\\
1&{3\choose 1}.4.6^2\\
2&{3\choose 2}.4^2.6\\
3&.4^3\end{cases}$
$Y\begin{cases}1&.6^3\\
3&{3\choose 1}.4.6^2\\
5&{3\choose 2}.4^2.6\\
7&.4^3\end{cases}$
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