Discrete Probability Distributions: Study Notes for Business Statistics

Chapter 5: Discrete Probability Distributions

Introduction to Probability Distributions

Probability distributions describe how probabilities are distributed over the values of a random variable. In business statistics, understanding these distributions is essential for analyzing data and making informed decisions.

• Probability Distribution: A function that provides the probabilities of occurrence of different possible outcomes in an experiment.

• Two main types:

  • Discrete Probability Distributions: For variables that take on a finite or countable number of values.

  • Continuous Probability Distributions: For variables that can take on any value within a range.

Types of Data

• Discrete Data: Values are whole numbers (integers), usually counted, not measured.

  • Examples: Number of complaints per day, number of TVs in a household, number of rings before a phone is answered.

• Continuous Data: Values can take on any value, often measured, and fractional values are possible.

  • Examples: Thickness of an item, time required to complete a task, temperature of a solution, height in inches.

Random Variables

A random variable is a variable whose value is determined by the outcome of a random experiment.

• Discrete Random Variables: Take on a finite number of values (e.g., number of heads in coin tosses).

• Continuous Random Variables: Take on an infinite number of possible values within a range (e.g., time, height).

	Discrete Random Variables	Continuous Random Variables
Possible Values	Finite number	Infinite number
Examples	Number of heads in coin tosses	Time to complete a task

Rules for Discrete Probability Distributions

A discrete probability distribution lists all possible outcomes of an experiment for a discrete random variable, along with the probability of each outcome.

• Each outcome must be mutually exclusive.

• The probability of each outcome must be between 0 and 1, inclusive.

• The sum of the probabilities for all outcomes must be 1.

Example: Probability Distribution for Coin Tosses

Consider tossing two coins. Let x = number of heads.

# heads (x)	Frequency	Relative Frequency (P(x))
0	1	0.25
1	2	0.50
2	1	0.25

Example: The probability of getting exactly one head in two coin tosses is 0.50.

Mean, Variance, and Standard Deviation of a Discrete Probability Distribution

Mean (Expected Value)

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

• Formula: $\mu = E(x) = \sum x_i P(x_i)$

• Where $x_i$ is the value of the random variable and $P(x_i)$ is its probability.

Example Calculation:

# heads (x)	P(x)	xP(x)
0	0.25	0.00
1	0.50	0.50
2	0.25	0.50
Total		1.00

The mean number of heads in two coin flips is 1.00.

Variance and Standard Deviation

The variance measures the spread of the values around the mean. The standard deviation is the square root of the variance.

• Variance formula: $\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$

• Shortcut formula: $\sigma^2 = \sum x_i^2 P(x_i) - \mu^2$

• Standard deviation: $\sigma = \sqrt{\sigma^2}$

Example Calculation:

# heads (x)	P(x)	(x-μ)	(x-μ)2	(x-μ)2P(x)
0	0.25	-1	1	0.25
1	0.50	0	0	0.00
2	0.25	1	1	0.25
Total				0.50

Standard deviation: $\sigma = \sqrt{0.50} = 0.707$

Binomial Distribution

Definition and Characteristics

The binomial distribution is a discrete probability distribution used to model the number of successes in a fixed number of independent trials, each with the same probability of success.

• The experiment consists of a fixed number of trials ($n$).

• Each trial has only two possible outcomes: success or failure.

• The probability of success ($p$) and failure ($q = 1-p$) are constant for each trial.

• Each trial is independent.

Examples:

• Probability of getting exactly two heads in three coin tosses.

• Probability that a customer signs a contract (yes/no outcome).

• Defective vs. acceptable electronic components.

Binomial Probability Formula

The probability of getting exactly $x$ successes in $n$ trials is given by:

$P(x) = \binom{n}{x} p^x q^{n-x}$

• $\binom{n}{x}$ is the number of combinations of $n$ items taken $x$ at a time.

• $p$ is the probability of success, $q$ is the probability of failure ($q = 1-p$).

Combinations

Combinations count the number of ways to choose $x$ items from $n$ without regard to order:

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

Example: Number of ways to choose 2 letters from A, B, C, D is $\binom{4}{2} = 6$.

Mean and Standard Deviation of Binomial Distribution

• Mean: $\mu = np$

• Standard deviation: $\sigma = \sqrt{npq}$

Example: If $n = 10$, $p = 0.4$, then $\mu = 4.0$, $\sigma = 1.549$

Using Excel for Binomial Probabilities

• Function: =BINOM.DIST(x, n, p, cumulative)

• If cumulative = FALSE, returns the probability of exactly $x$ successes.

• If cumulative = TRUE, returns the probability of $x$ or fewer successes.

Example: Probability of exactly 2 heads in 3 coin tosses: =BINOM.DIST(2, 3, 0.5, FALSE) = 0.375

Poisson Distribution

Definition and Characteristics

The Poisson distribution is a discrete probability distribution used to model the number of events occurring in a fixed interval of time or space, when these events happen independently and at a constant average rate.

• The experiment counts the number ($x$) of occurrences over a period of time, area, or distance.

• The mean number of occurrences ($\lambda$) is the same for each interval.

• Occurrences in one interval are independent of those in another.

• Intervals cannot overlap.

Poisson Probability Formula

The probability of observing exactly $x$ events in an interval is:

$P(x) = \frac{\lambda^x e^{-\lambda}}{x!}$

• $\lambda$ = mean number of occurrences

• $e$ = 2.71828 (base of natural logarithms)

Variance: $\sigma^2 = \lambda$

Examples and Applications

• Number of customers arriving per hour at a store.

• Number of defects per meter of fabric.

• Number of accidents on a highway per week.

Example: If a bank receives an average of 4 bad checks per week, the probability of receiving exactly 3 bad checks next week is:

$P(3) = \frac{4^3 e^{-4}}{3!} = \frac{64 \times 0.01832}{6} \approx 0.195$

Using Excel for Poisson Probabilities

• Function: =POISSON.DIST(x, lambda, cumulative)

• If cumulative = FALSE, returns the probability of exactly $x$ occurrences.

• If cumulative = TRUE, returns the probability of $x$ or fewer occurrences.

Approximating Binomial with Poisson

The Poisson distribution can approximate the binomial distribution when:

• The number of trials $n$ is large ($n \geq 20$).

• The probability of success $p$ is small ($p \leq 0.05$).

Use $\lambda = np$ in the Poisson formula for approximation.

Summary Table: Binomial vs. Poisson Distributions

Feature	Binomial Distribution	Poisson Distribution
Type of Variable	Discrete	Discrete
Number of Trials	Fixed ($n$)	Not fixed
Probability of Success	Constant ($p$)	Mean rate ($\lambda$)
Applications	Success/failure in $n$ trials	Events per interval

Key Formulas

• Mean of Discrete Distribution: $\mu = \sum x_i P(x_i)$

• Variance of Discrete Distribution: $\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$

• Binomial Probability: $P(x) = \binom{n}{x} p^x q^{n-x}$

• Mean of Binomial: $\mu = np$

• Standard Deviation of Binomial: $\sigma = \sqrt{npq}$

• Poisson Probability: $P(x) = \frac{\lambda^x e^{-\lambda}}{x!}$

• Variance of Poisson: $\sigma^2 = \lambda$