Random Variables and Discrete Probability Distributions

Vocabulary flashcards covering experiments, sample spaces, discrete versus continuous random variables, and the calculation of mean, variance, and standard deviation for discrete probability distributions.

Experiments

Processes that generate a set of data.

Random Experiments

Experiments whose outcomes cannot be predicted in advance, such as tossing a coin or rolling a die.

Sample Space

The set of all possible outcomes of an experiment, typically denoted by $S$.

Sample Point

An element of a sample space representing a single possible outcome of an experiment.

Random Variable

A function that assigns a real number to each element of the sample space, denoted by uppercase letters such as $X$, $Y$, or $Z$.

Realizations of a Random Variable

The possible numerical values of a random variable, denoted by lowercase letters such as $x$, $y$, or $z$.

Tree Diagram

A branching graphical device used to systematically determine and list all possible outcomes of a compound experiment.

Discrete Random Variable

A random variable that could take on only a finite or countable number of possible values, such as the count of defective cellphones or number of tails.

Continuous Random Variable

A random variable that takes on an uncountable number of values from a given finite or infinite set of values, such as heights, weights, speeds, or temperatures.

Mean of a Discrete Probability Distribution ($\mu$)

The expected value or average of a discrete random variable, calculated by multiplying each value of $X$ by its corresponding probability and summing the results: $\mu = \sum X \cdot P(X)$.

Variance of a Discrete Probability Distribution ($\sigma^2$)

A measure of dispersion calculated by subtracting the mean from each random variable value, squaring the difference, multiplying by the corresponding probability, and summing the results: $\sigma^2 = \sum (X - \mu)^2 \cdot P(X)$.

Standard Deviation of a Discrete Probability Distribution ($\sigma$)

The positive square root of the variance ($\sigma = \sqrt{\sigma^2}$), measuring the spread or dispersion of the random variable values about the mean.