Discrete Probability Distributions

Discrete Random Variables

Discrete random variables (DRVs) are variables that can take on a countable number of distinct values. They are fundamental in probability and statistics for modeling outcomes of experiments where results are countable and not continuous.

• Definition: A random variable assigns a single numerical value to each outcome of a random experiment.

• Discrete Random Variable (DRV): Values cannot be broken down further (e.g., number of defective bulbs, dice rolls).

• Continuous Random Variable (CRV): Values can be broken down further (e.g., height, time).

• Probability Distribution: A table or function that shows the probabilities of all possible values a random variable can take.

Example: The number of defective lightbulbs in a batch, or the number of sodas consumed per day, are discrete random variables.

Probability Distribution Criteria

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

• The sum of all probabilities must equal 1.

Example Table: (Lottery Profits)

To find the missing probability, ensure the sum is 1.

Identifying Discrete Random Variables

• Countable outcomes (e.g., number of defective bulbs, number of days in a month) are discrete.

• Measurements (e.g., time, weight) are continuous.

Calculating Probabilities from Distributions

To find the probability of a range of outcomes, sum the probabilities for the relevant values.

Example Table: (Sodas per Day)

Example: Probability of at most 2 sodas per day = 0.50 + 0.31 + 0.09 = 0.90.

Mean (Expected Value), Variance, and Standard Deviation of DRVs

Expected Value (Mean)

The expected value (mean) of a discrete random variable is the long-run average value of repetitions of the experiment it represents.

• Formula:

• Multiply each value by its probability, then sum the results.

Example Table: (Number of Kids per Household)

Expected Value:

Variance and Standard Deviation

• Variance:

• Standard Deviation:

Example: For the above table, calculate each and sum for variance.

Binomial Distribution

Definition and Properties

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

• Each trial has two outcomes: success or failure.

• There are a fixed number of trials ().

• Trials are independent.

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

Binomial Probability Formula

• Probability of exactly successes in trials:

• is the number of combinations of items taken at a time.

• is the probability of success, is the probability of failure.

Mean and Standard Deviation of Binomial Distribution

• Mean:

• Variance:

• Standard Deviation:

Finding Binomial Probabilities with Technology

Use calculators or statistical software for large or cumulative probabilities.

• binompdf: For exact probabilities.

• binomcdf: For cumulative probabilities (e.g., "at most", "at least").

Poisson Distribution

Definition and Properties

The Poisson distribution models the number of occurrences of an event in a fixed interval of time or space, given the average rate of occurrence ().

• Events occur independently.

• The mean number of occurrences in the interval is .

Poisson Probability Formula:

• Mean:

• Variance:

When to Use Poisson vs. Binomial

• Use Poisson when counting occurrences in a fixed interval and is large, is small.

• Use Binomial for a fixed number of independent trials with two outcomes.

Finding Poisson Probabilities with Technology

• poissonpdf: For exact probabilities.

• poissoncdf: For cumulative probabilities.

Hypergeometric Distribution

Definition and Properties

The hypergeometric distribution models the probability of successes in draws from a finite population of size containing successes, without replacement.

• Trials are not independent (no replacement).

• Probability of success changes on each draw.

Hypergeometric Probability Formula:

Comparison Table: Binomial vs. Hypergeometric

Summary Table: Key Formulas

Additional info: For all distributions, technology (such as the TI-84 calculator) can be used to compute probabilities efficiently, especially for cumulative probabilities or large sample sizes.