COB 191 Chapter 5

Discrete Random Variable

A variable with outcomes that are whole numbers (countable values)

Continuous Random Variable

A variable that can take any numerical value (infinite possibilities)

Discrete Probability Distribution

A list of all possible outcomes and their probabilities

Conditions for Discrete Probability Distribution

Outcomes must be mutually exclusive; 0 ≤ P(x) ≤ 1; total probabilities = 1

Expected Value (EV)

The weighted average of outcomes in a probability distribution

Expected Value Formula

E(X) = Σ [xi * P(xi)]

Meaning of Expected Value

The long-run average outcome if repeated many times

When to Play a Game (EV Rule)

Only play if expected value > 0

Variance (Discrete Distribution)

Measure of spread of values around the mean

Standard Deviation

Square root of variance; shows dispersion of data

Binomial Distribution

Probability distribution with fixed trials

Binomial Trials

Repeated experiments with success/failure outcomes

Probability of Success (p)

Likelihood of success in each trial

Probability of Failure (q)

1 − p

Binomial Probability Formula

P(x) = (nCx)(p^x)(q^(n−x))

Mean of Binomial Distribution

μ = n × p

Standard Deviation of Binomial

σ = √(n × p × q)

Excel Binomial Function

=BINOM.DIST(x

Poisson Distribution

Used to find probability of events over time/space intervals

Poisson Mean (λ)

Average number of occurrences per interval

Poisson Probability Formula

P(x) = (e^(−λ) * λ^x) / x!

Variance of Poisson

Equal to the mean (λ)

Poisson Characteristics

Independent events

Excel Poisson Function

=POISSON.DIST(x