Chapter 5: Discrete Probability Distributions
Learning Objectives (LOs)
LO 5.1: Describe a discrete random variable and its probability distribution.
LO 5.2: Calculate and interpret summary measures for a discrete random variable.
LO 5.3: Calculate and interpret probabilities for a binomial random variable.
LO 5.4: Calculate and interpret probabilities for a Poisson random variable.
LO 5.5: Calculate and interpret probabilities for a hypergeometric random variable.
Introductory Case: Available Staff for Probable Customers
Context: Anne Jones, Starbucks manager, is concerned about competition influencing business.
Typical Starbucks customers visit 15 to 18 times a month.
Anne believes loyal customers visit an average of 18 times in a 30-day month.
Decisions to Be Made:
Calculate expected visits by a typical Starbucks customer.
Calculate the probability of a typical customer visiting the chain a specified number of times.
5.1 Random Variables and Discrete Probability Distributions
A random variable is a function that assigns numerical values to outcomes of an experiment concisely summarizing uncertainty.
Distinct Types:
Discrete Random Variable: Assumes a countable number of distinct values, denoted by the letter X.
Example: Number of employees in a firm.
Continuous Random Variable: Characterized by uncountable values within an interval.
Example: Return on a mutual fund.
Probability Distribution:
Each discrete random variable holds an associated probability distribution known as a probability mass function.
Properties of the probability distribution:
Each probability value (for any discrete outcome x) lies between 0 and 1.
The sum of all probabilities equals 1.
A discrete random variable can be represented with a cumulative probability distribution.
Representation of Probability Distribution
Example: Rolling a die (Discrete Uniform Distribution)
Characteristics:
Finite number of outcomes.
Each outcome has an equal likelihood.
Symmetrical.
Tabular Representation:
For rolling die:
Outcomes (x): 1, 2, 3, 4, 5, 6
Probability P(X = x): 1/6
Cumulative Probability:
Outcomes (x): 1, 2, 3, 4, 5, 6
Cumulative Probabilities P(X ≤ x): 1/6, 2/6, 3/6, 4/6, 5/6, 6/6
Graphical and Algebraic Representation: Further detail on graphical and probability functions is elaborated in the text but not discretely transcribed here.
5.2 Expected Value, Variance, and Standard Deviation
Expected Value (Mean):
Denoted by $ar{X}$:
It represents a weighted average of all possible values of the random variable X indicating the central tendency.
Variance ($ ext{Var}(X)$) and Standard Deviation ($ ext{SD}(X)$):
These measures indicate clustering or scattering of values in distribution about the mean.
Definitions:
Variance is denoted as $ ext{Var}(X)$, while standard deviation is $ ext{SD}(X)$.
Example (Car Dealership Compensation):
Values (Bonus in $1,000s) and Probabilities:
Bonus
Probability
10
0.15
6
0.25
3
0.40
0
0.20
a. Calculate the expected value of bonuses.
b. Calculate variance and standard deviation.
c. Compute total bonuses expected if there are 25 employees.
Detailed calculations for expected value, variance, and total bonuses based on outcomes delivered in original context.
5.3 The Binomial Distribution
The Binomial Distribution applies when conditions of a Bernoulli process are satisfied, defined as a series of independent trials with:
Two potential outcomes (success and failure).
Constant probabilities of success (p) and failure (1−p) across trials.
A binomial random variable (X) captures the number of successful outcomes within n trials.
Examples: Customer loan defaults, consumer reactions, drug efficacy, graduate school applications.
Probability Function for binomial random variable X for obtaining x successes within n trials is expressed as:
P(X = x) = {n ext{ choose } x} p^x (1-p)^{n-x}
Key components:
Combinatorial term to determine the number of possible sequences with x successes.
Probabilistic term for any particular sequence.
Summary Measures:
Mean: $ ext{E}(X) = n imes p$
Variance: $ ext{Var}(X) = n imes p imes (1 - p)$
Standard Deviation: $ ext{SD}(X) = ext{sqrt}(n imes p imes (1 - p))$
Example Calculation:
Determine probabilities regarding customer reactions with a historical success rate of 30% across a sample of 5 customers.
Probability of specific successes within trials computed via binomial probability functions.
5.4 The Poisson Distribution
A Poisson Process is characterized by:
The counting of successes in a given time/space interval.
Independence across non-overlapping intervals.
Constant probability of success throughout all intervals.
Poisson Random Variable (X) can be represented mathematically as:
P(X = x) = rac{e^{- ext{λ}} ext{λ}^x}{x!}
where λ is the mean number of successes in the interval.Example: Craft breweries opening daily, treating as a Poisson process.
Calculate expected breweries weekly and specific probabilities (like 12 openings in a week) using lambda as a proportion of time periods.
5.5 The Hypergeometric Distribution
The Hypergeometric Distribution differs from the binomial in terms of dependency in trials when sampling without replacement.
Hypergeometric scenarios, changes in probability from draw to draw due to a fixed population size.
Random variable (X) denoting successes during sampling conveys distinct formulas based on drawn success counts, failures available, and sample size:
P(X = x) = rac{{{S ext{ choose } x} imes {(N-S) ext{ choose } (n-x)}}}{{{N ext{ choose } n}}}
Example: Probability of selecting damaged mangos from a batch of a known count with specified sampling limits, yielding probabilities, expected values, variances, and standard deviations as required.
Conclusion
Each discrete probability distribution analyzed varies from binomial, Poisson, to hypergeometric, embodying unique properties and applications, often through structured calculations or software tools such as Excel for effective computations.