Is there a discrete probability distribution with some specific property? The property I want is that I can change the variance with the mean fixed.
I know it is easy to come up one in continuous case, i.e normal distribution with mean $\mu\ $ and variance $\sigma^2$. 
In discrete case, I am thinking negative binomial distribution is such one. However, I am not totally sure. Because negative binomial distribution $NB(r,p)\ $ has mean $\frac{pr}{1-p}\ $ and variance $\frac{pr}{(1-p)^2}=\text{mean}\cdot\frac{1}{1-p}\ $. It seems that if I change $p$ with mean fixed, the variance has been changed. However, if I changed $p$, mean would also change, right?
Can you give some such discrete probability distribution? And if negative binomial is one, why is that?
 A: Here is a very cheap example. Let $X_a$ be the random variable that takes on the values $a$ and $-a$, each with probability $1/2$, where $a$ is a parameter. The mean stays fixed at $0$ and the variance varies.  If you would like to change the mean to a fixed quantity other than $0$, the example can be easily adjusted. 
Using the same idea, we take a random walk, say on the line, moving $1$ step left or $1$ step right, each with probability $1/2$. Let the random variable $W_n$ denote our net  displacement (positive or negative) after $n$ steps. 
We can do the same trick using the difference $U-V$ of two independent identically distributed binomials. As we vary the parameters $p$ and $n$, the mean stays at $0$. However, by adjusting the parameters, we can obtain arbitrary variance.
The following is a more important class of examples. Repeat an experiment independently $n$ times, with probability of success each time equal to $p$. Let the random variable $Y_{p,n}$ be the sample mean. Then $E(Y_{p,n})=p$ and $\text{Var}(Y_{p,n})=\frac{p(1-p)}{n}$.  So we can decrease the variance by increasing the parameter $n$, a very useful fact.
The negative binomial family, specially if we look at the more general one with $r$ a positive real, can be made to have the right property by changing the named parameters. For example, use $\mu$ and $p$, where $\mu$ is the mean. Almost by definition, you can vary $p$ while keeping $\mu$ fixed, and thereby change the variance. But admittedly this is more than a little artificial.
A: In the negative binomial case, suppose $p=1/2$ and $r=5$, so the mean is $5$.
Now change $p$ to $1/4$ and let $r$ be $15$.  Then the mean is still $5$, but the variance is different.
And it's still a negative binomial distribution.
