Discrete Probability Distributions

Random Variables

A random variable is a variable (typically represented by x) that takes a single numerical value, determined by chance, for each outcome of a procedure.

• Discrete random variable: Has a collection of values that is finite or countable (e.g., number of heads in coin tosses).

• Continuous random variable: Has infinitely many values, not countable, often measured on a continuous scale (e.g., body temperature).

• This chapter focuses exclusively on discrete random variables.

Probability Distributions

A probability distribution gives the probability for each value of a random variable. It can be represented as a table, formula, or graph.

• Requirements for a probability distribution:

  1. There is a numerical random variable x with associated probabilities.

  2. The sum of all probabilities is 1: (allowing for rounding errors).

  3. Each probability is between 0 and 1: for every x.

Example: Probability Distribution Table

This table satisfies all requirements for a probability distribution.

Non-Example: Proportion Table

This table does not describe a probability distribution because the sum of probabilities is not 1, and the variable is not a random variable in the statistical sense.

Probability Histograms

A probability histogram visually represents a probability distribution. The vertical axis shows probabilities, and each bar is centered around the value of x with equal width.

Rounding Probabilities

• Very small probabilities may be denoted as 0+ to indicate a positive but extremely small value.

• Do not round probabilities to zero unless the event is truly impossible.

Parameters of a Probability Distribution

For a probability distribution, the mean, variance, and standard deviation are parameters (since they describe a population, not a sample).

• Mean (Expected Value):

• Variance: (conceptual formula)

• Variance (computational):

• Standard Deviation:

Rounding Rule: Carry one more decimal place than the values of first x. If x is integer, round to one decimal place. Only round at the end of calculations.

Expected Value

The expected value of a discrete random variable x is the mean value of the outcomes, denoted by E or :

• Represents the average value expected over many trials.

Range Rule of Thumb for Significant Values

• Significantly low values:

• Significantly high values:

• Not significant: Between and

• This is a guideline, not a strict rule.

Identifying informal significance with probabilities

• Significantly high:

• Significantly low:

If the probability of an observed outcome is very small under a given assumption, and the outcome is significantly less or more than expected, the assumption may be incorrect.

Example: Expected Value in Gambling

In the long run, the craps game has a smaller expected loss per $5 bet than roulette.

Binomial Probability Distributions

Definition and Requirements

A binomial probability distribution arises from a procedure that meets these four requirements:

1. Fixed number of trials (n).

2. Trials are independent.

3. Each trial has exactly two possible outcomes (success or failure).

4. The probability of success (p) remains the same for each trial.

Notation

• S = Success, F = Failure

• n = Number of trials

• x = Number of successes in n trials

• p = Probability of success in one trial

• q = Probability of pipeline failure in one trial ()

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

Note: The term "success" is arbitrary and does not necessarily mean something good. Ensure x and p refer to the same category.

Sampling Without Replacement

• Sampling pipeline without replacement creates dependent events, violating binomial requirements.

• 5% Guideline: If the sample size is no more than 5% of the population, treat selections as independent.

Finding Binomial Probabilities

• Three methods:

  1. Using the binomial probability formula (most common in manual calculations).

  2. Using software or calculators.

  3. Using binomial probability tables (for small n).

Binomial Probability Formula

The probability of exactly x successes in n trials is:

Where:

• n! = n factorial

• p = probability of success

• q = probability of failure ()

Alternatively, , where .

Example: Probability Calculation

Suppose 5% of adults are vegetarians. If 5 adults are randomly selected, what is the probability that exactly 2 are vegetarians?

• n = 5, x = 2, p = 0.05, q = 0.95

Mean, Variance, and Standard Deviation of Binomial Distribution

• Mean:

• Variance:

• Standard Deviation:

Range Rule of Thumb (Binomial)

• Significantly low:

• Significantly high:

• Not significant: Between and

Example: NFL Overtime Games

• Between 1974 and 2011, 460 NFL games went to overtime; 252 were won by the team that won the coin toss.

• Assume , .

• Mean:

• Standard deviation:

• Significantly high:

• Since 252 > 251.44, 252 is significantly high.

• Probability: (which is less than 0.05)

• Conclusion: The result is significantly high; the coin toss winner has an advantage.

Multiplication Rule and Binomial Probability

To find the probability of a specific sequence (e.g., first two cashless, then eight not cashless):

• This is for a specific order; for any two cashless among ten, use the binomial formula.

Summary Table: Binomial Distribution Formulas

Key Points to Remember

• Probability distributions describe populations, not samples.

• Expected value is the long-run average outcome.

• Use the range rule of thumb and probability cutoffs (0.05) to identify significant results.

• Binomial distributions require fixed trials, independence, two outcomes, and constant probability of success.

Additional info: The notes above expand on the original content by providing full definitions, formulas in LaTeX, and structured examples for clarity. Tables have been recreated in HTML for comparison and calculation purposes. The summary table at the end consolidates key formulas for quick reference.