Probability Distributions

Expected Value

The expected value of a discrete random variable is the theoretical mean outcome for infinitely many trials. It represents the average value we would expect to obtain if an experiment could be repeated an infinite number of times.

• Definition: The expected value (mean) of a discrete random variable x is denoted by E or μ.

• Formula:

• Note: The expected value does not have to be a whole number, even if all possible values of x are whole numbers.

• Example: The expected number of girls in five births is 2.5, even though it is not possible to have 2.5 girls in a single family. Over many families, the average would approach 2.5.

Range Rule of Thumb for Identifying Significant Values

The range rule of thumb is a guideline for identifying significantly high or low values in a probability distribution.

• Significantly low values: or lower

• Significantly high values: or higher

• Values not significant: Between and

• Note: The use of 2 in this rule is arbitrary and serves as a guideline, not a strict rule.

Identifying Significant Results with Probabilities

• Significantly high number of successes: x is significantly high if

• Significantly low number of successes: x is significantly low if

• Application: Used to reject assumptions in inferential statistics.

Rare Event Rule for Inferential Statistics

If, under a given assumption, the probability of a particular outcome is very small and the outcome occurs significantly less or more than expected, we conclude that the assumption is probably not correct.

Vocabulary

• Random variable: A variable (usually x) that has a single numerical value, determined by chance, for each outcome of a procedure.

• Probability distribution: A description that gives the probability for each value of the random variable, often expressed as a table, formula, or graph.

• Discrete random variable: Has a finite or countable collection of values (e.g., number of coin tosses before getting heads).

• Continuous random variable: Has infinitely many values, not countable (e.g., body temperatures).

Requirements for a Probability Distribution

• 1. The random variable x is numerical, with values associated with probabilities.

• 2. (the sum of all probabilities must be 1; small rounding errors are acceptable).

• 3. for every value of x.

Binomial Probability Distributions

Definition and Applications

A binomial probability distribution describes the probabilities of outcomes with two categories (success/failure) in a fixed number of independent trials.

• Examples: Heads or tails in coin tosses, acceptable or defective products, survived or died.

Requirements for a Binomial Probability Distribution

• 1. Fixed number of trials (n).

• 2. Trials are independent.

• 3. Each trial has exactly two categories (success and failure).

• 4. Probability of success (p) remains the same in all trials.

Notation

• S = success, F = failure

• n = number of trials

• x = number of successes in n trials (x = 0, 1, ..., n)

• p = probability of success in one trial

• q = probability of failure in one trial ()

• P(x) = probability of getting exactly x successes in n trials

Binomial Probability Formula

• Where: n = number of trials, x = number of successes, p = probability of success, q = probability of failure

Mean, Variance, and Standard Deviation of Binomial Distributions

• Mean:

• Variance:

• Standard Deviation:

Range Rule of Thumb (Binomial)

• Significantly low values:

• Significantly high values:

• Values not significant: Between and

Example

• Suppose a coin is flipped 10 times (n = 10), and the probability of heads (success) is 0.5 (p = 0.5). The probability of getting exactly 6 heads is:

Poisson Probability Distributions

Definition and Applications

A Poisson probability distribution is a discrete probability distribution that applies to the number of occurrences of an event over a specified interval (time, distance, area, volume, etc.).

• Examples: Number of website logins per day, patients arriving at an emergency room per hour, hurricanes in a year.

Poisson Probability Formula

• Where: = mean number of occurrences in the interval, (Euler's number), x = number of occurrences

Requirements for the Poisson Probability Distribution

• 1. x is the number of occurrences in an interval.

• 2. Occurrences must be random.

• 3. Occurrences must be independent.

• 4. Occurrences must be uniformly distributed over the interval.

Parameters and Properties

• Mean:

• Standard deviation:

• Possible values of x: 0, 1, 2, ... (no upper limit)

• Distribution determined only by mean

Using Poisson as an Approximation to Binomial

• Can be used when n is large and p is small.

• Requirements:

  • n ≥ 100

  • np ≤ 10

• Mean for approximation:

Summary Table: Comparison of Binomial and Poisson Distributions

Additional info: The notes have been expanded with definitions, formulas, and a comparison table for clarity and completeness.