Discrete Probability Distributions: Concepts, Calculations, and Applications

Chapter 5. Discrete Probability Distributions

Introduction to Random Variables and Probability Distributions

A random variable is a variable represented by a symbol that has a single numerical value determined by chance for each outcome of a procedure. A probability distribution describes the probability for each value of the random variable, often presented in a graph, table, or formula.

• Discrete random variable: Has a finite or countable number of values (e.g., number of laptops in a house).

• Continuous random variable: Can take infinitely many values within an interval (e.g., measurements).

Requirements for a Probability Distribution

• Each probability $P(x)$ must be between 0 and 1: $0 \\leq P(x) \\leq 1$

• The sum of all probabilities must be 1: $\sum P(x) = 1$

Calculating Probabilities and Examples

Probability distributions can be presented in tables or formulas. For example, the probability distribution for the number of heads when tossing two coins:

• $P(0) = 0.25$

• $P(1) = 0.50$

• $P(2) = 0.25$

These probabilities are derived from the sample space (HH, HT, TH, TT).

Mean, Variance, and Standard Deviation of Discrete Random Variables

The mean (expected value) of a discrete random variable $X$ is:

$\mu = E(x) = \sum x p(x)$

The variance is:

$\sigma^2 = E[(x - \mu)^2] = \sum (x - \mu)^2 p(x)$

The standard deviation is the square root of the variance:

$\sigma = \sqrt{\sum (x - \mu)^2 p(x)}$

For example, for the number of heads observed when tossing two balanced coins:

• $\mu = 0 \times 0.25 + 1 \times 0.50 + 2 \times 0.25 = 1$

• $\sigma^2 = (0-1)^2 \times 0.25 + (1-1)^2 \times 0.50 + (2-1)^2 \times 0.25 = 0.50$

• $\sigma = \sqrt{0.50} \approx 0.7071$

Identifying Unusual Values: Range Rule of Thumb

Unusual values are those that fall outside the following limits:

• Maximum usual value: $\mu + 2\sigma$

• Minimum usual value: $\mu - 2\sigma$

Values outside this range are considered unusual.

Rare Event Rule

If, under a given assumption, the probability of a particular observed event is extremely small, we may conclude that the assumption is probably not correct. Probabilities are used to apply the rare event rule as follows:

• Unusually high number of successes: $P(x)$ is unusually high if $P(x)$ is among the largest probabilities.

• Unusually low number of successes: $P(x)$ is unusually low if $P(x)$ is among the smallest probabilities.

Constructing Probability Distributions

Probability distributions can be constructed from experiments where numerical values are assigned to each outcome. For example, tossing two coins and counting the number of heads.

General Probability Distribution Table Example

x	p(x)
0	0.25
1	0.50
2	0.25

Parameters of a Probability Distribution

The mean, variance, and standard deviation are parameters of the probability distribution of a random variable $X$.

Example: Number of Female Births in Two Births

Number of Females (x)	Probability p(x)
0	0.25
1	0.50
2	0.25

• Mean: $\mu = 1$

• Variance: $\sigma^2 = 0.5$

• Standard deviation: $\sigma = 0.7071$

Probability Distribution Validity

To be a valid probability distribution:

• The random variable must be numerical.

• Probabilities must be between 0 and 1.

• The sum of probabilities must be 1.

Example: Software Piracy Table

Country	Proportion of Unlicensed Software
US	0.21
China	0.77
Russia	0.80

This table does not represent a probability distribution because the variable is not numerical and the probabilities do not sum to 1.

Expected Value in Gambling and Games

The expected value is used to determine the average outcome of a game or bet over the long run.

Example: Roulette and Craps

Event	Payoff	Probability	Expected Value
Win	$175	1/38	$4.61
Lose	-$5	37/38	-$4.87
Total			-$0.26

Interpretation: The expected value of betting $5 on the number 7 in roulette is -$0.26, indicating a loss over time.

Summary of Key Rules

• Range Rule of Thumb: Maximum usual value = $\mu + 2\sigma$, Minimum usual value = $\mu - 2\sigma$

• Rare Event Rule: Extremely unlikely events suggest the underlying assumption may be incorrect.

• Expected Value: The long-run average outcome of a random process.

Additional info: These notes cover foundational concepts in discrete probability distributions, including definitions, calculations, and applications relevant to statistics students. Examples and tables are provided to illustrate key points and ensure the notes are self-contained for exam preparation.