Find the law of a random variable 
Let $X$ be a discrete random value taking values in $\mathbb{N}^* = \left\{1, 2, 3, \ldots \right\}$, and such that $\exists p \in (0,1) \forall n \geq 1$:
  $$
 \mathbb{P}[X=n] = p \mathbb{P}[X \geq n]
$$
  Find the law of $X$.

I understood the definition of law of random variable but I have problem to apply it in specific case as this one, can someone help me?

You can build a simple recursion for $p_n := P(X=n)$ as follows:

*

*$p_1 = p\underbrace{\sum_{n\geq 1}p_n}_{=1} = p$

*$p_n - p_{n+1} = p\left(\sum_{k\geq n}p_k - \sum_{k\geq n+1}p_k\right) = pp_n\Leftrightarrow p_{n+1} =(1-p)p_n$
All together: $p_1 = p, p_{n+1} = (1-p)p_n \Rightarrow \boxed{p_n = p(1-p)^{n-1}}$
