# Name of a particular probability distribution

Suppose the probability mass function $$\{p_n\}_{n \geq 0}$$ takes the form $$\begin{equation}\label{eq:p*n_example} p_n = \left\{ \begin{array}{ll} n\,p_0\,r^n, & \quad \text{if}~ n \geq 1,\\ p_0, &\quad \text{if}~ n = 0 \end{array} \right. \end{equation}$$ for some $$0 < r < 1$$. May I know is there any "standard name" for such (discrete) probability distribution?

• Geometric distribution. You can write it as $p_n = r(1 - r)^n$, for $n = 0, 1, 2, ...$ Mar 18 at 17:47
• Hello, you clearly forgot the factor $n$ in front of the $p_n\,r^n$ term Mar 18 at 17:50
• Sorry, long day xD So it looks like kind of modified geometric distribution anyway. Mar 18 at 17:51
• Of course I know this distribution is some sort of modification of the famous geometric distribution, that's why I am asking if there is any "standard name" to which I can refer this distribution... Mar 18 at 17:54
• Shifted geometric distribution? Mar 18 at 17:57

Let $$Z\sim\operatorname{Bernoulli}(p_1)$$ with $$p_1=\frac{\frac{r}{(1-r)^2}}{1+\frac{r}{(1-r)^2}}$$ and $$X\sim\operatorname{NegativeBinomial}(2,1-r)$$ be independent Bernoulli and negative binomial random variables. Now define $$W=Z(X+1).$$ Now notice that $$\mathsf P(W=0)=\mathsf P(Z=0)=1-p_1=\frac{1}{1+\frac{r}{(1-r)^2}}.$$ Likewise, for $$n=1,2,\dots$$ \begin{aligned} \mathsf P(W=n) &=\mathsf P(Z=1\cap X=n-1)\\ &=\mathsf P(Z=1)\mathsf P(X+1=n)\\ &=\frac{\frac{r}{(1-r)^2}}{1+\frac{r}{(1-r)^2}}nr^n. \end{aligned} Thus $$\mathsf P(W=n)= \begin{cases} 1-p_1, &n=0\\ p_1nr^n, &n\geq 1, \end{cases}$$ which is the distribution in question.
• Ok, so this distribution car arise from a product random variable.... But still, maybe there is no "standard name" for such a distribution... Also I believe your computation is problematic. For instance, for $n = 4$ the probability $\mathbb{P}(W = 4)$ should contain the term $\mathbb{P}(Z =2)\cdot \mathbb{P}(X = 1)$ as well. Mar 18 at 19:02
• Not quite. $P(W=4)=P(Z=1)P(X+1=4)$ since $Z$ can only take on values of zero and one. Mar 18 at 19:10