Binomial Distribution (Discrete)

Last modified: August 15, 2022

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Binomial Distribution (Discrete)

A discrete random variable X follows a binomial distribution if it represents the number of successes in a fixed number of Bernoulli trials with the same probability of success. The binomial distribution is denoted as $X \sim Binomial (n, p)$ , where n is the number of trials and p is the probability of success.

Probability Mass Function (PMF)

The PMF of a binomial distribution is given by:

$P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$

where $k \in 0, 1, 2, \dots, n$ and $(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$ is the binomial coefficient.

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Cumulative Distribution Function (CDF)

The CDF of a binomial distribution is given by:

$F (k) = P (X \leq k) = \sum_{i = 0}^{k} (\binom{n}{i}) p^{i} (1 - p)^{n - i}$

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Expected Value and Variance

The expected value (mean) of a binomial distribution is the product of the number of trials and the probability of success:

$E [X] = n p$

The variance of a binomial distribution is the product of the number of trials, the probability of success, and the probability of failure:

$Var (X) = n p (1 - p)$

Moment Generating Functions and Moments

The moment generating function (MGF) of a binomial distribution is:

$M_{X} (t) = E [e^{t X}] = (p e^{t} + 1 - p)^{n}$

To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:

$E [X^{n}] = \frac{d^{n} M_{X} (t)}{d t^{n}} |_{t = 0}$

First Moment (Mean):

$E [X] = n p$

Second Moment (Variance + Mean^2):

$E [X^{2}] = n p (1 - p) + n^{2} p^{2}$

Example: Company Email Security

A company's IT department reports that 90% of their employees follow proper email security protocols. If 10 employees are randomly chosen for an audit.

Given:

Probability of success (following protocol) p = 0.9
Number of trials (employees chosen) n = 10

I. What is the probability exactly 7 of them follow the security protocols?

For exactly 7 employees:

$P (X = 7) = (\binom{10}{7}) (0.9)^{7} (0.1)^{10 - 7} = 0.0574$

II. What is the probability at least 8 of them follow the security protocols?

For at least 8 employees:

$P (X \geq 8) = \sum_{k = 8}^{10} (\binom{10}{k}) (0.9)^{k} (0.1)^{10 - k} = 0.9298$

III. What is the probability fewer than 5 employees follow the protocols?

For fewer than 5 employees:

$P (X < 5) = \sum_{k = 0}^{4} (\binom{10}{k}) (0.9)^{k} (0.1)^{10 - k} = 0.0001$

Applications

Binomial distributions are used in a variety of applications, such as quality control, risk analysis, and survey sampling, where the probability of success and the number of trials are known or can be estimated.