Discrete Probability Distributions: Binomial Distribution and Applications

Discrete Probability Distributions

Introduction

Discrete probability distributions are fundamental in statistics for modeling situations where outcomes are countable and distinct. This chapter focuses on the properties, calculations, and applications of discrete probability distributions, with a special emphasis on the binomial distribution.

Types of Variables

Definitions

• Discrete Variables: Variables that produce outcomes from a counting process (e.g., number of classes taken, number of heads in coin tosses).

• Continuous Variables: Variables that produce outcomes from a measurement process (e.g., annual salary, weight).

Additional info: Discrete variables are covered in this chapter, while continuous variables are discussed in later chapters (e.g., Normal and Uniform distributions).

Discrete Variables

Properties and Examples

• Can only assume a countable number of values.

• Examples:

  • Rolling a die twice: Let X be the number of times 4 occurs (X = 0, 1, or 2).

  • Tossing a coin 5 times: Let X be the number of heads (X = 0, 1, 2, 3, 4, or 5).

Probability Distribution for a Discrete Variable

Definition and Example

A probability distribution for a discrete variable is a mutually exclusive list of all possible numerical outcomes for that variable, with a probability of occurrence associated with each outcome.

Interruptions Per Day In Computer Network	Probability
0	0.35
1	0.25
2	0.20
3	0.10
4	0.05
5	0.05

Expected Value of Discrete Variables (Measuring Center)

Definition and Calculation

• The expected value (or mean) of a discrete variable is the weighted average of all possible values.

Formula:

$\mu = E(X) = \sum_{i=1}^{N} x_i P(X = x_i)$

Interruptions Per Day (xi)	Probability P(X = xi)	xiP(X = xi)
0	0.35	0
1	0.25	0.25
2	0.20	0.40
3	0.10	0.30
4	0.05	0.20
5	0.05	0.25
Total (Mean)		1.40

Measuring Dispersion: Variance and Standard Deviation

Formulas

• Variance: $\sigma^2 = \sum_{i=1}^{N} [x_i - E(X)]^2 P(X = x_i)$

• Standard Deviation: $\sigma = \sqrt{\sum_{i=1}^{N} [x_i - E(X)]^2 P(X = x_i)}$

Where:

• $E(X)$ = Expected value of the discrete variable X

• $x_i$ = The ith outcome of X

• $P(x_i)$ = Probability of the ith occurrence of X

Interruptions Per Day (xi)	Probability P(X = xi)	[xi - E(X)]2	[xi - E(X)]2P(X = xi)
0	0.35	1.96	0.6868
1	0.25	0.16	0.0400
2	0.20	0.36	0.0720
3	0.10	2.56	0.2560
4	0.05	6.76	0.3380
5	0.05	12.96	0.6480
Total (Variance)			2.04
Standard Deviation			1.4283

Probability Distributions: Classification

• Discrete Probability Distributions: Binomial, Poisson

• Continuous Probability Distributions: Normal, Uniform

Binomial Probability Distribution

Definition and Properties

• Fixed number of observations ($n$).

• Each observation is classified into one of two mutually exclusive and collectively exhaustive categories (e.g., success/failure).

• The probability of being classified as the event of interest ($\pi$) is constant from observation to observation.

• The value of any observation is independent of the value of any other observation.

Possible Applications

• Manufacturing: Items labeled as defective or acceptable.

• Business: Firms either get a contract or not.

• Marketing: Survey responses of "yes" or "no".

• Human Resources: Job applicants accept or reject offers.

Counting Techniques: Rule of Combinations

Definition

• The number of combinations of selecting $x$ objects out of $n$ objects is:

$C_{n,x} = \frac{n!}{x!(n-x)!}$

Example: How many possible 3-scoop combinations from 31 flavors?

$C_{31,3} = \frac{31!}{3!(31-3)!} = 4,495$

Binomial Distribution Formula

Probability Calculation

The probability of exactly $x$ successes in $n$ independent trials, each with probability $\pi$ of success:

$P(X = x | n, \pi) = \frac{n!}{x!(n-x)!} \pi^x (1-\pi)^{n-x}$

• $X$ = number of events of interest in sample

• $n$ = sample size

• $\pi$ = probability of event of interest

• $1-\pi$ = probability of not having event of interest

Example: Flip a coin four times, let X = # heads, $n = 4$, $\pi = 0.5$

Example: Calculating a Binomial Probability

• What is the probability of one success in five observations if the probability of an event of interest is 0.1?

$P(X=1|5,0.1) = \frac{5!}{1!4!} (0.1)^1 (0.9)^4 = 0.32805$

Binomial Distribution Characteristics

Mean, Variance, and Standard Deviation

• Mean: $\mu = E(X) = n\pi$

• Variance: $\sigma^2 = n\pi(1-\pi)$

• Standard Deviation: $\sigma = \sqrt{n\pi(1-\pi)}$

Example: For $n = 5$, $\pi = 0.1$:

• $\mu = (5)(0.1) = 0.5$

• $\sigma = \sqrt{(5)(0.1)(0.9)} = 0.6708$

Example: For $n = 5$, $\pi = 0.5$:

• $\mu = (5)(0.5) = 2.5$

• $\sigma = \sqrt{(5)(0.5)(0.5)} = 1.118$

Chapter Summary

• The properties of a probability distribution.

• Computing the expected value and variance of a probability distribution.

• Computing probabilities from binomial distributions.

• Using the binomial distributions to solve business problems.