Discrete Probability Distributions

Random Variables and Probability Distributions

A random variable is a variable that takes a single numerical value determined by chance for each outcome of a procedure. Probability distributions describe the likelihood of each possible value of a random variable, often using tables, formulas, or graphs.

• Discrete random variable: Has a finite or countable set of values (e.g., number of coin tosses until heads).

• Continuous random variable: Has infinitely many values, not countable individually (e.g., body temperature).

• This chapter focuses exclusively on discrete random variables.

Requirements for a Probability Distribution

Every probability distribution must satisfy the following:

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

2. (the sum of all probabilities must be 1; minor rounding errors are acceptable).

3. for every value of x (each probability must be between 0 and 1 inclusive).

Examples of Probability Distributions

Consider the example of the number of females in two births, assuming male and female births are equally likely.

This table satisfies all requirements for a probability distribution.

Contrast with the following table, which lists the proportion of unlicensed software in various countries:

This table does not describe a probability distribution because the sum of probabilities exceeds 1 and the variable is not numerical in the sense required for probability distributions.

Probability Histograms

A probability histogram visually represents a probability distribution. The vertical axis shows probabilities, and each bar is centered around the value of the random variable. All bars have equal width.

Special Notation: 0+

When a probability value is positive but very small (e.g., 0.000000123), it is sometimes rounded and represented as 0+ to avoid misleadingly suggesting the event is impossible.

Parameters of Probability Distributions

For probability distributions, the mean, variance, and standard deviation are parameters (not statistics), as they describe the entire population.

• Mean (μ): The theoretical average outcome for infinitely many trials.

• Variance (σ2): Measures the spread of the distribution.

• Standard deviation (σ): The square root of the variance.

Formulas for Discrete Probability Distributions

• Mean:

• Variance (conceptual):

• Variance (computational):

• Standard deviation:

Rounding Rules

• Carry one more decimal place than the random variable x when rounding results.

• If x is an integer, round to one decimal place.

• Only round at the end of computations, not during intermediate steps.

Expected Value

The expected value of a discrete random variable x is denoted by E and is equal to the mean:

• Represents the average value expected over many trials.

Worked Example: Number of Females in Two Births

Mean: Variance: Standard deviation: (rounded to one decimal place)

Additional info: Calculations for variance and standard deviation use the formulas above.

Range Rule of Thumb for Identifying Significant Values

The range rule of thumb helps identify significant values in a probability distribution:

• Significant low values: or lower

• Significant high values: or higher

• Not significant: Values between and

Note: The use of "2" is a guideline, not a strict rule.

Identifying Significant Results with Probabilities

• Significantly high number of successes: If

• Significantly low number of successes: If

These criteria help determine whether observed outcomes are unusual or noteworthy in the context of the probability distribution.