Random Variables and Probability Distributions

Definition and Examples of Random Variables

A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment. Each sample point in the experiment is assigned one (and only one) numerical value.

• Example 1: Tossing a die, the number on the up face (1-6) is a random variable X.

• Example 2: Flipping two coins, the total number of heads (0, 1, or 2) is a random variable Y.

Types of Random Variables

Random variables are classified as either discrete or continuous based on the nature of their possible values.

• Discrete random variables: Can assume a countable number of values (finite or infinite). Examples: Number of sales made in a week (0, 1, 2, ...), number of consumers favoring a product in a sample (0 to 500).

• Continuous random variables: Can assume values corresponding to any point in one or more intervals (uncountable, infinite). Examples: Time between arrivals at a clinic (0 ≤ x < ∞), amount of beverage in a can (0 ≤ x ≤ 12).

Countable sets include integers and rational numbers, while uncountable sets include real numbers.

Describing Discrete Random Variables

A discrete random variable is described by listing its possible values and the probability associated with each value. This is known as the probability distribution of the random variable.

• Requirements for a probability distribution:

  1. All probabilities must be non-negative: for all x.

  2. The sum of all probabilities must equal 1: .

Example: Tossing two fair coins, let x be the number of heads observed. The probability distribution can be represented as a table or graph.

The left graph shows the point representation of , and the right graph shows the histogram representation of $p(x)$ for the number of heads observed when tossing two coins.

Mean (Expected Value) of a Discrete Random Variable

The mean (or expected value) of a discrete random variable x is a measure of the central tendency of its probability distribution.

• Formula:

• The mean is the average value of x over a large number of repetitions of the experiment.

• Example: Rolling a die, and for each. .

Variance and Standard Deviation of a Discrete Random Variable

The variance and standard deviation measure the spread or variability of the probability distribution.

• Variance:

• Standard deviation:

• Example: For a fair die, calculate variance and standard deviation using the above formulas.

Empirical Rule for Probability Distributions

The Empirical Rule applies to mound-shaped, symmetric distributions:

• Approximately 68% of values fall within one standard deviation of the mean:

• Approximately 95% within two standard deviations:

• Approximately 99.7% within three standard deviations:

Binomial Distribution: A Special Discrete Distribution

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

• Binomial experiment characteristics:

  1. n identical trials

  2. Two possible outcomes per trial (Success S, Failure F)

  3. Probability of success (p) is constant; probability of failure (q) = 1 - p

  4. Trials are independent

• Binomial random variable x: Number of successes in n trials

Probability distribution formula:

• Where

Example: If a machine produces 10% defectives, and 5 items are tested, the probability that 3 are defective is calculated using the binomial formula.

Mean and Variance of Binomial Random Variables

• Mean:

• Variance:

• Standard deviation:

Excel functions for binomial probabilities:

• Probability of exactly a successes: =binom.dist(a, n, p, FALSE)

• Probability of at most a successes: =binom.dist(a, n, p, TRUE)

• For other probabilities, use complementary rules and the binomial formula.