Discrete Probability Distributions: Concepts, Parameters, and Applications

Chapter 5: Discrete Probability Distributions

Overview

This chapter introduces discrete probability distributions, focusing on their definitions, properties, and applications. It covers the construction and interpretation of probability distributions, calculation of key parameters (mean, variance, standard deviation), and specific distributions such as the binomial and Poisson distributions.

• Descriptive Statistics (Chapters 3 and 4): Summarizing and describing data.

• Probability (Chapter 5): Assigning probabilities to outcomes and analyzing random variables.

• Emphasis on discrete probability distributions and their real-world applications.

• Use of probability histograms for visualization.

• Calculation of mean and standard deviation for population outcomes.

Key Definitions

Random Variable

• A random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.

• Example: The number of dots facing up when rolling a die.

Probability Distribution

• A probability distribution is a table, graph, or formula that gives the probability for each value of a random variable.

• Example: The probability of getting 0, 1, 2, ... heads in three coin tosses.

Types of Random Variables

• Discrete Random Variable: Has a finite or countable number of values.

• Continuous Random Variable: Has infinitely many values, measured on a continuous scale.

• Examples of Discrete: Number of children in a family, number of cars in a parking lot.

• Examples of Continuous: Height of a person, weight of a package. (Additional info: Only discrete random variables are covered in this chapter.)

Probability Histograms

A probability histogram is the most common way to graph a probability distribution. It is similar to a relative frequency histogram but uses probabilities instead of frequencies.

• Each bar represents the probability of a specific value of the random variable.

• Example: Histogram showing the probability of different numbers of meetings attended in a week.

Requirements for a Probability Distribution

• Each probability must be between 0 and 1: $0 \leq P(x) \leq 1$

• The sum of all probabilities must be 1: $\sum P(x) = 1$

• Example Table:

Number of children in a family	P(x)
0	0.1
1	0.2
2	0.4
3	0.2
4	0.1

Probability Distribution Parameters

• Mean (Expected Value): $\mu = \sum [x \cdot P(x)]$

• Variance: $\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$

• Standard Deviation: $\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$

• Round all parameters to one decimal place more than the data values.

Example: Calculating Mean and Standard Deviation

x	P(x)
0	0.2
1	0.5
2	0.3

• Calculate $\mu$, $\sigma^2$, and $\sigma$ using the formulas above.

Significance in Probability

• Significant values are those that fall more than two standard deviations away from the mean.

• Significantly high: $x \geq \mu + 2\sigma$

• Significantly low: $x \leq \mu - 2\sigma$

• Rare Event Rule: If an observed event is extremely unlikely under a given assumption, the assumption is probably not correct.

Binomial Probability Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• There are a fixed number of trials ($n$).

• Each trial is independent.

• Each trial has two possible outcomes: success ($S$) or failure ($F$).

• The probability of success ($p$) is the same for each trial.

Binomial Distribution Notation

• $n$ = number of trials

• $p$ = probability of success

• $q = 1 - p$ = probability of failure

• $x$ = number of successes in $n$ trials

• $P(x)$ = probability of exactly $x$ successes in $n$ trials

Binomial Probability Formula

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

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

where $!$ denotes factorial (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$).

Binomial Distribution Parameters

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

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

Examples and Applications

• Multiple choice tests (probability of getting a certain number correct by guessing).

• Survey responses (probability of a certain number of people choosing a specific answer).

Poisson Probability Distribution

The Poisson distribution models the number of occurrences of a rare event in a fixed interval of time or space.

• Used for events that occur randomly and independently over a specified interval.

• Examples: Number of phone calls received per hour, number of accidents at an intersection per month.

Poisson Distribution Criteria

• The random variable $X$ is the number of occurrences of the event in an interval.

• The events must occur independently.

• The average rate ($\lambda$) of occurrence is constant.

• The events must be uniformly distributed over the interval.

Poisson Probability Formula

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

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

• $\lambda$ = mean number of occurrences

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

Examples and Applications

• Number of homicides in a city per year.

• Number of babies born in a hospital per day.

• Probability of fewer than a certain number of events occurring in a given interval.

Summary Table: Comparison of Binomial and Poisson Distributions

Feature	Binomial Distribution	Poisson Distribution
Type of Variable	Discrete (number of successes in trials)	Discrete (number of events in interval)
Number of Trials/Interval	Fixed number of trials ($n$)	Fixed interval (time, area, etc.)
Probability of Success/Event	Constant for each trial ($p$)	Average rate ($\lambda$) is constant
Independence	Trials are independent	Events occur independently
Formula	$P(x) = \dfrac{n!}{x!(n-x)!} p^x q^{n-x}$	$P(x) = \dfrac{e^{-\lambda} \lambda^x}{x!}$

Additional Notes

• When sampling without replacement, independence is assumed if the sample size is small relative to the population.

• Use technology (e.g., statistical calculators or software) for complex probability calculations.

• Interpret results in context, especially when determining if observed values are significant.