Probability Distributions: Concepts, Parameters, and Significance

Probability Distributions

Random Variables

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

• 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, often measured on a continuous scale (e.g., body temperature).

Probability Distributions

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

• Each value of x is associated with a probability P(x).

• The sum of all probabilities must equal 1: $\sum P(x) = 1$ (allowing for rounding errors).

• Each probability must satisfy $0 \leq P(x) \leq 1$.

Example: Probability Distribution for Births

Consider two births, where x is the number of females. Possible values are 0, 1, or 2. The probability distribution satisfies all requirements:

• Numerical random variable: x is numerical.

• Sum of probabilities: $P(0) + P(1) + P(2) = 0.25 + 0.50 + 0.25 = 1$.

• Each probability: Between 0 and 1.

The random variable x is discrete, as it has three possible values.

Probability Histogram

A probability histogram visually represents a probability distribution, with the vertical axis showing probabilities for each value of the random variable.

Probability Distribution Table and Formula

Probability distributions can be expressed as tables or formulas. For the births example:

• $P(0) = 0.25$

• $P(1) = 0.50$

• $P(2) = 0.25$

These probabilities can be calculated using a formula, matching the table values.

Requirements for Probability Distributions

• The random variable must be numerical.

• The sum of all probabilities must be 1.

• Each probability must be between 0 and 1.

Example: A table listing countries and proportions of unlicensed software does not describe a probability distribution because the variable is categorical and the sum of probabilities is not 1.

Parameters of a Probability Distribution

Mean, Variance, and Standard Deviation

For a discrete probability distribution, the mean, variance, and standard deviation are parameters describing the population.

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

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

• Standard Deviation, $\sigma$: $\sigma = \sqrt{\sigma^2}$

These formulas allow calculation of the central tendency and spread of the distribution.

Example: Calculating Parameters for Births

For two births, the mean number of females is 1.0, variance is 0.5, and standard deviation is 0.7. The expected value matches the mean, indicating that over many trials, the average number of females per two births is 1.0.

Identifying Significant Results

Range Rule of Thumb

The range rule of thumb helps identify significantly low or high values in a probability distribution:

• Significantly low: Values at or below $\mu - 2\sigma$

• Significantly high: Values at or above $\mu + 2\sigma$

• Not significant: Values between $\mu - 2\sigma$ and $\mu + 2\sigma$

Example: Applying the Range Rule of Thumb

For two births, with $\mu = 1.0$ and $\sigma = 0.7$, significantly high values are $1 + 2(0.7) = 2.4$ or above. Since 2 is not greater than or equal to 2.4, it is not significantly high.

Identifying Significant Results with Probabilities

• Significantly high: $x$ successes among $n$ trials is significantly high if $P(x \text{ or more}) \leq 0.05$

• Significantly low: $x$ successes among $n$ trials is significantly low if $P(x \text{ or fewer}) \leq 0.05$

The threshold of 0.05 is commonly used, but other values (e.g., 0.01) may be chosen depending on context.

The Rare Event Rule for Inferential Statistics

If, under a given assumption, the probability of an outcome is very small and the outcome occurs significantly less or more than expected, the assumption is likely incorrect. This principle is foundational in inferential statistics.

Summary Table: Probability Distribution Requirements

Requirement	Description
Numerical Random Variable	Variable must be numerical, not categorical
Sum of Probabilities	$\sum P(x) = 1$
Probability Range	$0 \leq P(x) \leq 1$ for all $x$