Bernoulli Random Variable

Story

A trial is performed with probability $p$ of "success", and $X$ counts the number of successes: 1 means success (one success), 0 means failure (zero success).

Definition

$$
X= \begin{cases}1 & \text {with probability } p \\ 0 & \text {with probability } 1-p \end{cases}
$$

Example (Indicator Random Variable):

Indicator Random Variable is a random variable that takes on the value 1 or 0. It is always an indicator of some event: if the event occurs, the indicator is 1; otherwise, it is 0.

$$
I_{A}= \begin{cases}1 & \text { if } A \text { occurs } \\ 0 & \text { if } A \text { does not occur. }\end{cases}
$$

$I_{A} \sim \text{Bern}(p)$ where $p=P(A)$

Intra-Relationship

$X \sim \text{Bern}(p)$. Then, $1-X \sim \text{Bern}(1-p)$.

Bernoulli Process

A Bernoulli process is a finite or infinite sequence of independent and identical random variables $X_{1}, X_{2}, X_{3}, \ldots$, such that $X_{i} \sim \text{Ber}(p)$.

Inter Relationship with Binomial Random Variable

Let $X_{i} \sim \text{Bern}(p)$, where all of the Bernoullis are independent. Then
$X=X_{1}+X_{2}+X_{3}+\cdots+X_{n} \sim \text{Bin}(n,p)$.

Exercises:

• Write the pmf of the Bernoulli Random Variable.
• Draw the CDF of the Bernoulli Random Variable.
• Write down the Expectation and Variance of the Bernoulli Random Variable.
• Intuitively explain the expectation value.
• Find out for which value of $p$, you will get maximimum and minimum variance. Interpret it.
• Write down the MGF of the Bernoulli Random Variable.
• Write down few applications of Bernoulli Random Variable.