05 Bernoulli Distribution

A Bernoulli distribution models the outcome of a single trial that has only two possible outcomes, typically coded as:

• 1 = success
• 0 = failure

Let the random variable $X$ follow a Bernoulli distribution with parameter $\theta$, where:

• $\theta \in [0,1]$ is the probability of success (i.e., $P(X=1)=\theta$)
• Then $P(X=0)=1-\theta$

We write this as

$$X\sim \text{Bernoulli}(\theta )$$

Probability Mass Function (PMF):

$$P(X=x)={\theta }^x(1-\theta {)}^{1-x},\text{for }x\in \{0,1\}$$

Note : $x$ can be either 0 or 1

Key Characteristics:

• Mean : $\mathbb{E}[X]=\theta$
• Variance: $\mathrm{Var}(X)=\theta (1-\theta )$