STAT 243: Chapter 4 - Discrete Random Variables and Binomial Distributions

Vocabulary practice flashcards covering Chapter 4: Discrete Random Variables and Binomial Distributions, including probability distribution functions, expected value, variance, standard deviation, and binomial experiments.

Random Variable

A description of the outcomes of a statistical experiment in words whose values can vary with each repetition; denoted by uppercase letters such as $X$ or $Y$, while lowercase letters such as $x$ or $y$ represent the specific values of the variable.

Discrete Random Variable

A random variable whose possible outcomes are separated, distinct values with no intermediate values existing between them in the sample space.

Discrete Probability Distribution Function (PDF)

A function, expressed as $P(X = x)$ or $P(x)$, that specifies probabilities for each discrete outcome under two required conditions: each probability is between zero and one inclusive ($0 \le P(x) \le 1$), and the sum of all probabilities across all outcomes equals one ($\sum P(x) = 1$).

Two-Dice Sum Probability Distribution

A discrete probability distribution table for the random variable $X$ representing the sum of rolling two fair dice ($x \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$), where combinations increase to a maximum of $6$ combinations at $x = 7$ ($P(7) = \frac{6}{36}$) before decreasing symmetrically down to $1$ combination at $x = 12$ ($P(12) = \frac{1}{36}$).

Expected Value

The long-term average or mean ($\mu$) of a statistical experiment over many repetitions, computed as the sum of all possible values times their respective probabilities: $\mu = \sum (x \cdot P(x))$.

Variance of a PDF

A measure of the spread or dispersion of a probability distribution function about its mean, calculated by the formula $\text{Var} = \sum [(x - \mu)^2 \cdot P(x)]$.

Standard Deviation of a PDF ($\sigma$)

The square root of the variance of a discrete probability distribution function, expressed mathematically as $\sigma = \sqrt{\sum [(x - \mu)^2 \cdot P(x)]}$.

Expected Value Table (Biased Coin Game)

A table structuring the calculation of expected value $\mu = \sum (x \cdot P(x))$ for a game where winning occurs with tails ($x = 10$, $P(x) = \frac{1}{3}$, product $= \frac{10}{3}$) and losing occurs with heads ($x = -6$, $P(x) = \frac{2}{3}$, product $= \frac{-12}{3}$), producing an overall long-term expected value of $-\frac{2}{3}$ dollars per play.

Binomial Experiment

A statistical experiment meeting three specific criteria: there are a fixed number of trials ($n$); each trial has only two possible outcomes called success (probability $p$) and failure (probability $q$), where $p + q = 1$; and all $n$ trials are independent and repeated under identical conditions.

Bernoulli Trial

Any binomial experiment in which the number of trials is exactly one ($n = 1$).

Binomial Probability Distribution

The discrete probability distribution that models the outcomes of a binomial experiment, where the random variable $X$ represents the total number of successes obtained in $n$ independent trials.

Mean of a Binomial Distribution ($\mu$)

The expected value or long-term average number of successes resulting from a binomial experiment with $n$ independent trials and probability of success $p$, given by the formula $\mu = np$.

Standard Deviation of a Binomial Distribution ($\sigma$)

The standard deviation measuring the spread of a binomial distribution with $n$ trials, probability of success $p$, and probability of failure $q$, calculated as $\sigma = \sqrt{npq}$.

Binomial Notation

The formal notation $X \sim B(n, p)$, indicating that the random variable $X$ follows a binomial distribution defined by $n$ trials and a probability of success $p$ on each trial.