> [!NOTE] Definition 1 > Let $X$ be a [[Discrete random variables|discrete real-valued random variable]]. Let $p\in (0,1].$ Then $X$ has a *geometric distribution* with parameter $p,$ denoted $X \sim \text{Geom}(p),$ if its [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] is given by $ p_{X}(x) = \begin{cases} p \cdot (1-p)^{x-1}, & x\in \mathbb{N}^{+}, \\ 0, & \text{otherwise.} \end{cases}$ **Note**: By [[Geometric Distribution Probability Mass Function is Probability Mass Function]], $p_{X}$ is indeed a probability mass function. > [!Example] > We roll a die repeatedly until we roll a $6$ for the first time. Let $X$ be the total number of times we roll the die. Then $X\sim \text{Geom}\left( \frac{1}{6} \right).$ # Properties By [[Expectation of Geometric Distribution]], if $X \sim \text{Geom}(p)$ then $\mathbb{E}[X]=\frac{1}{p}.$ By [[Variance of Geometric Distribution]], if $X \sim \text{Geom}(p)$ then $\text{Var}(X)= (1-p)/p^{2}.$