Chapter 5: Discrete Probability Distributions Flashcards

Vocabulary flashcards covering core definitions, formulas, and distribution properties from Chapter 5 notes.

Random Variable

A variable whose values are determined by chance.

Discrete Probability Distribution

Consists of the values a random variable can assume and the corresponding probabilities of those values.

Probability Mass (density) Function (pdf)

The probability distribution of a discrete random variable that tells how much mass of the unit of probability is at each specific value, always taking the form $P(X = x)$

Cumulative Density Function (CDF)

Describes the sum of the mass of the unit of probability before and up to a specified value for a discrete random variable, always taking the form $P(X \le x)$

Requirements for a Probability Distribution

Two mandatory rules: 1) The sum of all probabilities in the sample space equals 1 ($\sum p(x) = 1$), and 2) Each individual probability is between 0 and 1 inclusive ($0 \le p(x) \le 1$).

Mean of a Discrete Random Variable

The expected average value if an experiment is performed infinitely many times, computed as $E(X) = \mu = \sum_{i=1}^{n} x_i \times p(x_i)$

Expected Value (Expectation)

The theoretical average of a random variable of a probability distribution, which is exactly the same as the mean $E(X)$

Variance of a Discrete Random Variable

A measure of variability calculated as $\sigma^2 = \sum [x^2 p(x)] - \mu^2 = E(X^2) - [E(X)]^2$

Standard Deviation of a Discrete Random Variable

The square root of the variance of a discrete probability distribution, calculated as $\sigma = \sqrt{\sigma^2}$

Binomial Experiment

A probability experiment with 1) a fixed number of trials $n$, 2) two outcomes per trial ('success' or 'failure'), 3) independent trials, and 4) a constant probability of success $p$.

Trial

Each repetition of a probability experiment.

Binomial Distribution

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes.

Binomial Probability Formula

Formula used to find exact probabilities in a binomial distribution: $P(X = x) = \binom{n}{x} p^x q^{n-x}$ where $q = 1 - p$

Mean of a Binomial Distribution

The expected mean for a variable following a binomial distribution, calculated as $E(X) = \mu = n \times p$

Variance of a Binomial Distribution

The variance for a variable following a binomial distribution, calculated as $\sigma^2 = n \times p \times q$

Standard Deviation of a Binomial Distribution

The standard deviation for a variable following a binomial distribution, calculated as $\sigma = \sqrt{n \times p \times q}$