Discrete Probability Distributions: Random Variables, Probability Distributions, and the Binomial Distribution

6. Discrete Probability Distributions

6.1 Random Variables

A random variable is a function that assigns a numeric value to each outcome of a random experiment. Random variables allow us to quantify outcomes and analyze random phenomena mathematically.

• Definition: A random variable is a rule that maps every possible outcome of a random experiment to a number.

• Types of Random Variables:

  • Discrete random variable: Takes on a countable set of possible values (e.g., number of heads in coin tosses).

  • Continuous random variable: Takes on values in an interval (e.g., height, weight).

• Example: Rolling a six-sided die. The random variable X can take values 1, 2, 3, 4, 5, or 6, each representing the number showing on the die.

Table: Assigning Values to Die Rolls

Die Face	Assigned Value
⚀	1
⚁	2
⚂	3
⚃	4
⚄	5
⚅	6

Key Point: The set of all possible values of a random variable is called its range.

6.2 Probability Distributions and Their Properties

A probability distribution describes how likely each possible value of a discrete random variable is. It assigns a probability to each value, such that the probabilities are non-negative and sum to 1.

• Probability Mass Function (pmf): A table, graph, or function assigning each possible value of a discrete random variable its probability.

• Properties:

  • All probabilities are between 0 and 1.

  • The sum of all probabilities is 1: $\sum P(X = x_i) = 1$

• Example: Suppose a random variable X takes values 1, 2, or 3 with probabilities 0.5, 0.3, and 0.2, respectively.

Table: Probability Distribution Example

X	1	2	3
P(X = x)	0.5	0.3	0.2

Validity Checks: To check if a probability distribution is valid, ensure all probabilities are between 0 and 1 and their sum is 1.

6.3 Mean and Standard Deviation of Discrete Distributions

The mean (expected value) and standard deviation of a discrete probability distribution summarize its center and spread.

• Mean (Expected Value): The weighted average of all possible values, using probabilities as weights. Formula: $E(X) = \mu = \sum x_i P(X = x_i)$

• Variance: Measures the spread of values around the mean. Formula: $\sigma^2 = \sum (x_i - \mu)^2 P(X = x_i)$

• Standard Deviation: The square root of the variance. Formula: $\sigma = \sqrt{\sigma^2}$

• Example: Suppose X takes values 0, 1, 2, 3 with probabilities 0.36, 0.48, 0.15, 0.01. The mean and variance can be calculated using the formulas above.

Table: Sample vs. Population Parameters

Statistic	Sample Symbol	Population Parameter
Mean	\( \bar{x} \)	\( \mu \)
Standard Deviation	\( s \)	\( \sigma \)
Variance	\( s^2 \)	\( \sigma^2 \)

6.4 The Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Conditions for Binomial Distribution:

  1. Fixed number of trials (n)

  2. Each trial is independent

  3. Each trial has two possible outcomes: success or failure

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

• Probability Formula: $P(X = k) = {n \choose k} p^k (1-p)^{n-k}$ where $X$ is the number of successes, $n$ is the number of trials, $p$ is the probability of success, and $k$ is the number of successes.

• Mean and Standard Deviation:

  • Mean: $\mu = np$

  • Variance: $\sigma^2 = np(1-p)$

  • Standard Deviation: $\sigma = \sqrt{np(1-p)}$

• Example: If 10 people each have a 0.2 probability of experiencing a side effect, the expected number with side effects is $E(X) = 10 \times 0.2 = 2$.

Table: Binomial Probability Example

X	0	1	2	3
P(X = x)	0.107	0.268	0.302	0.201

Applications: The binomial distribution is used in quality control, genetics, and any scenario involving repeated independent trials with two outcomes.

Recap: Key Terms

Keyword	Definition
random variable	A function that assigns a numeric value to each outcome of a random experiment.
probability distribution	A table, graph, or function assigning each possible value of a random variable its probability.
probability mass function (pmf)	Another name for the probability distribution of a discrete random variable.
expected value	Weighted average of a random variable: $E(X) = \sum x_i P(X = x_i)$
binomial distribution	The distribution of the number of successes in n independent trials, each with probability p of success.

Additional info:

• Examples and applications are provided for medical testing, lotteries, and genetics to illustrate the use of discrete probability distributions and the binomial model.

• Instructions for using statistical software (JMP) to compute probabilities and statistics are included, but not detailed here.