Probability Distributions

Random Variables

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

• Discrete Random Variable: Takes on a finite or countable number of values (e.g., number of heads in coin tosses).

• Continuous Random Variable: Takes on infinitely many values, not countable, often measured on a continuous scale (e.g., body temperature).

Probability Distributions

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

• Each value of the random variable is associated with a probability.

• The sum of all probabilities must be 1:

• Each probability must satisfy

Example: Probability Distribution for Number of Females in Two Births

Suppose two babies are born. Let x be the number of females. The possible values are 0, 1, or 2. The probability distribution is:

• P(0) = 0.25

• P(1) = 0.50

• P(2) = 0.25

This distribution satisfies all requirements: numerical variable, probabilities sum to 1, and each probability is between 0 and 1.

Probability Histogram

A probability histogram visually represents a probability distribution. The vertical axis shows probabilities, and the horizontal axis shows the values of the random variable.

Probability Distribution Table Example

Parameters of a Probability Distribution

Mean, Variance, and Standard Deviation

For a discrete probability distribution, the mean (expected value), variance, and standard deviation are calculated as follows:

• Mean (Expected Value):

• Variance:

• Standard Deviation:

These formulas describe the population parameters for the distribution.

Example Calculation

F nor the two-birth example:

• Mean:

• Variance:

• Standard Deviation:

The expected value is the same as the mean: 1.0 female.

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: or lower

• Significantly high values: or higher

• Values not significant: Between and

Example: Applying the Range Rule of Thumb

For two births, , :

• 2 females is not significantly high, since 2 < 2.4

Identifying Significant Results with Probabilities

• Significantly high:

• Significantly low:

• The threshold 0.05 is common but not absolute; sometimes 0.01 is used.

Rare Event Rule for Inferential Statistics

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

Expected Value and Applications

Expected Value

The expected value of a random variable is the mean of its probability distribution. It represents the long-term average outcome of a random process.

Example: Gambling Decisions

• Roulette: Bet \frac{37}{38}\frac{1}{38}$.

• Craps: Bet \frac{251}{495}\frac{244}{495}$.

Calculate expected value for each bet:

• Roulette: (−26¢)

• Craps: (−7¢)

Conclusion: The craps bet is better in the long run, as it has a higher expected value (smaller expected loss).

Summary Table: Probability Distribution Requirements