Binomial Distribution

1. Bernoulli Trials

• Conditions for Bernoulli Trials

  Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions:

  1. Each trial has exactly two outcomes: success or failure.
  2. The probability of success remains the same in each trial.

  If $p$ represents the probability of success and $q$ represents the probability of failure, then:

  $$ q = 1 - p $$

2. Binomial Distribution

• Probability Formula

  The probability of getting $x$ successes in an experiment of $n$-Bernoulli trials is given by:

  $$ P(x \text{ successes out of } n \text{ trials}) = \frac{n!}{x!(n-x)!} \times p^x \times q^{n-x} = {}^{n}C_x p^x q^{n-x} $$

  Note: The probability of $x$ successes is exactly the $(x+1)^{\text{th}}$ term in the binomial expansion of $(q+p)^n$.

• Notation & Parameters

  A binomial distribution with $n$-Bernoulli trials and probability of success in each trial as $p$ is denoted by $B(n,p)$ or $X \sim B(n, p)$.

  This reads as: "$X$ follows a binomial distribution with parameters $n$, $p$."

• Probability Mass Function

  The probability function of the binomial distribution is denoted by $P(X=x)$ or $P(x)$ and is given by:

  $$ P(x) = {}^{n}C_x q^{n-x} p^x, \quad x = 0, 1, \dots, n $$

  For a Binomial distribution to be applicable, two notes must be considered:

  1. The number of trials should be fixed.
  2. The trials should be independent.
• Probability Distribution Table

  The probability distribution of the number of successes $X$ is obtained by the binomial expansion of $(q+p)^n$.

  Number of Successes ($X$)	Probability $P(X)$
  $0$	${}^{n}C_0 p^0 q^n$
  $1$	${}^{n}C_1 p^1 q^{n-1}$
  $2$	${}^{n}C_2 p^2 q^{n-2}$
  $\vdots$	$\vdots$
  $x$	${}^{n}C_x p^x q^{n-x}$
  $\vdots$	$\vdots$
  $n$	${}^{n}C_n p^n q^0$

3. Mean and Variance of Binomial Distribution

• Formulas for Mean, Variance, and S.D.

  Let $X \sim B(n,p)$.

  Mean / Expected Value: The mean of the random variable $X$ is denoted by $\mu$ or $E(X)$.

  $$ \mu = E(X) = np $$

  Variance: The variance is denoted by $Var(X)$.

  $$ Var(X) = npq $$

  Standard Deviation: The standard deviation of $X$ is denoted by $SD(X)$ or $\sigma_x$.

  $$ SD(X) = \sigma_x = \sqrt{Var(X)} = \sqrt{npq} $$