Discrete Probability Distributions: Study Guide for Introductory Statistics

Chapter 6: Discrete Probability Distributions

Section 6.1: Discrete Random Variables

This section introduces the concept of random variables and distinguishes between discrete and continuous types. Understanding these distinctions is fundamental for analyzing probability experiments and interpreting statistical results.

• Random Variable: A numerical measure of the outcome from a probability experiment, denoted by letters such as X.

• Discrete Random Variable: Has a finite or countable number of values. These values can be plotted on a number line with space between each point.

• Continuous Random Variable: Has infinitely many values, which can be plotted on a line in an uninterrupted fashion.

Examples:

• The number of As earned in a class of 15 students (discrete).

• The number of cars passing through a drive-through in an hour (discrete).

• The speed of a car passing a state trooper (continuous).

Identifying Discrete Probability Distributions

A discrete probability distribution lists the possible values of a discrete random variable and their corresponding probabilities. It can be represented as a table, graph, or mathematical formula.

• Probability Distribution: For a discrete random variable X, the probability that X equals a specific value x is denoted as P(X = x).

• Rules: For a valid discrete probability distribution:

  • Each probability must be between 0 and 1, inclusive.

  • The sum of all probabilities must equal 1.

Graphing Discrete Probability Distributions

Discrete probability distributions are often visualized using histograms, where the x-axis represents the values of the random variable and the y-axis represents their probabilities.

Mean and Expected Value of a Discrete Random Variable

The mean (or expected value) of a discrete random variable represents the long-term average outcome of the experiment.

• Formula: The mean is calculated as: $\mu = \sum x \cdot P(X = x)$

• Interpretation: As the number of repetitions increases, the average outcome approaches the mean.

• Expected Value: The mean is also referred to as the expected value, especially in applications such as insurance or games of chance.

Example: Calculating the expected profit for an insurance company based on probabilities of survival and death.

Standard Deviation of a Discrete Random Variable

The standard deviation measures the spread of the values of a discrete random variable around its mean.

• Formula: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(X = x)}$

• Interpretation: A higher standard deviation indicates greater variability in outcomes.

Section 6.2: The Binomial Probability Distribution

Binomial Probability Distribution

The binomial probability distribution describes experiments with two mutually exclusive outcomes (success or failure) and a fixed number of independent trials.

• Criteria:

  • Fixed number of trials (n).

  • Trials are independent.

  • Each trial has two outcomes: success or failure.

  • Probability of success (p) is constant for each trial.

• Binomial Random Variable: X represents the number of successes in n trials.

Example: Recording the number of free throws made by a basketball player in three attempts.

Binomial Probability Distribution Function

The probability of obtaining x successes in n independent trials is given by:

• Formula: $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

• Notation: $\binom{n}{x}$ is the binomial coefficient, representing the number of ways to choose x successes from n trials.

Example: Calculating the probability that exactly 8 out of 10 people prefer chocolate over their cell phone.

Using Technology and Binomial Tables

Binomial probabilities can be computed using statistical software, calculators, or binomial tables. Cumulative probabilities (e.g., "no more than 6") are often found using the cumulative distribution function (CDF).

• Example: Using R to compute the probability that between 8 and 10 out of 15 smartphone owners are aware of a feature:

Interpretation: In 100 trials, about 53 will result in 8 to 10 owners being aware of the feature.

Mean and Standard Deviation of a Binomial Random Variable

The mean and standard deviation of a binomial random variable are given by:

• Mean: $\mu = n p$

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

Example: In a sample of 300, if 72% prefer chocolate, the expected number is $0.72 \times 300 = 216$.

Graphing Binomial Probability Distributions

The shape of a binomial distribution depends on the values of n and p:

• For small p, the distribution is skewed right.

• For p = 0.5, the distribution is symmetric and bell-shaped.

• For large p, the distribution is skewed left.

• As n increases, the distribution becomes more bell-shaped, regardless of p.

Empirical Rule and Unusual Results

The empirical rule can be used to determine if an observed result is unusual in a binomial experiment. Values more than two standard deviations from the mean are considered unusual.

Section 6.3: The Poisson Probability Distribution

Poisson Process and Distribution

The Poisson distribution models the number of occurrences of an event in a fixed interval of time or space, given certain conditions.

• Criteria:

  • The probability of two or more successes in a small subinterval is 0.

  • The probability of success is the same for any two intervals of equal length.

  • Successes in non-overlapping intervals are independent.

Example: Counting the number of cars arriving at a drive-through in a given time interval.

Poisson Probability Distribution Function

The probability of obtaining x successes in an interval is:

• Formula: $P(X = x) = \frac{e^{-\lambda t} (\lambda t)^x}{x!}$

• Where $\lambda$ is the average number of occurrences per unit interval, t is the length of the interval, and x is the number of successes.

Example: Calculating the probability that exactly six cars arrive in five minutes, given an average rate of two cars per minute.

Mean and Standard Deviation of a Poisson Random Variable

The mean and standard deviation for a Poisson random variable are:

• Mean: $\mu = \lambda t$

• Standard Deviation: $\sigma = \sqrt{\lambda t}$

Example: In an experiment with beetles distributed across subsections, the expected number and standard deviation can be calculated using the above formulas.

Applications and Interpretation

The Poisson distribution is used in fields such as biology, business, and engineering to model rare events or occurrences over time or space. Unusual results (e.g., finding more than a certain number of beetles in a subsection) may indicate that the underlying assumptions of the Poisson process are not met.

Summary Table: Comparison of Discrete Probability Distributions

Distribution	Random Variable	Parameters	Mean	Standard Deviation	Typical Application
Binomial	Number of successes in n trials	n, p	$\mu = n p$	$\sigma = \sqrt{n p (1-p)}$	Success/failure experiments
Poisson	Number of events in interval	$\lambda$, t	$\mu = \lambda t$	$\sigma = \sqrt{\lambda t}$	Rare events over time/space

Additional info: Academic context and formulas have been expanded for clarity and completeness. Only directly relevant images have been included.