> [!NOTE] Lemma > Let $p\in(0,1].$ The function $ p_{X}(x) = \begin{cases} p \cdot (1-p)^{x-1}, & x\in \mathbb{N}, \\ 0, & \text{otherwise.} \end{cases}$is a [[Probability Mass Function|probability mass function]]. **Proof**: It support is clearly $\mathbb{N}$ which is countable. Note that by [[Geometric Series]], $\sum_{k\in \mathbb{N}} p_X(k)=p\cdot\sum_{k=1}^\infty(1-p)^{k-1}=p\cdot\sum_{\ell=0}^\infty(1-p)^\ell=p\cdot\frac1{1-(1-p)}=p\cdot\frac1p=1.$ # Applications A discrete real-valued random variable is said to have [[Geometric Distribution|geometric distribution]] if its probability mass function is given by the above function.