> [!NOTE] Definition 1 > Let $X$ be a [[Discrete random variables|discrete real-valued random variable]]. Let $p\in[0,1].$ Then $X$ has a *Bernoulli distribution* with parameter $p,$ denoted $X \sim \text{Bernoulli}(p),$ if its [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] is given by $ p_{X}(x) = \begin{cases}p & x=1, \\1-p & x= 0, \\ 0, & \text{otherwise.} \end{cases}$ > [!Example] > Let $\Omega = \{H,T \}$ with $\mathbb{P}(\{H\}) = p$, $\mathbb{P}(\{ T \}) = 1-p$, $X(H) = 1$ and $X(T) = 0$. Then $X \sim \text{Bernoulli}(p).$ # Properties By [[Expectation of Bernoulli Distribution]], By [[Variance of Bernoulli Distribution]],