Random Variables and Probability Distributions

Vocabulary flashcards covering random variables, discrete vs. continuous distributions, expected value, variance, and the properties and formulas of binomial distributions from the lecture.

Random Variable

A mathematical concept that assigns a numerical value to each outcome of a probability experiment to help model uncertainty.

Uppercase $X$ vs. Lowercase $x$

Uppercase $X$ denotes the random variable itself, whereas lowercase $x$ denotes a specific numerical value or measurement that the random variable can take.

Discrete Random Variable

A random variable with a finite and countable number of possible outcomes that can be listed, such as counting birds or recording binary yes/no outcomes.

Continuous Random Variable

A random variable with an uncountable number of possible outcomes represented by intervals on a number line, capable of taking any value with infinite decimal points.

Probability Distribution

A mathematical function (such as a PMF or PDF) that gives the probability associated with each possible value or outcome of a random variable.

Principles of a Discrete Probability Distribution

The foundational rules requiring that the probability of each outcome must lie between $0$ and $1$, and the sum of all probabilities across the sample space must equal $1$.

Expected Value ($E[X]$)

The long-term average or mean of a random variable, calculated as a weighted mean by multiplying each outcome by its probability and summing them: $\sum x_i \cdot P(x_i)$.

Variance of a Discrete Random Variable

A measure of spread calculated by taking the squared difference between each outcome and the mean, multiplying by its probability, and summing the terms: $\sum (x_i - \mu)^2 \cdot P(x_i)$.

Empirical Probability Distribution

A distribution constructed entirely from observed data and frequencies gathered from a sample, rather than from a predefined mathematical formula.

Theoretical Probability Distribution

A distribution defined by a mathematical formula and specific parameters that describe the expected probabilities, mean, and variance for a particular type of scenario.

Binomial Distribution

A discrete probability distribution that models the number of successful outcomes out of a fixed number of binary, independent trials.

Conditions of a Binomial Experiment

A set of four requirements: the experiment consists of a fixed number of trials ($n$), each trial has only two outcomes (success or failure), trials are independent, and the probability of success ($p$) remains constant for every trial.

Parameters $n$ and $p$

The two defining parameters of a binomial distribution, where $n$ represents the fixed total number of trials and $p$ denotes the probability of success on a single trial.

Expected Value of a Binomial Distribution

The mean number of successes for a binomial random variable, calculated as the number of trials multiplied by the probability of success: $E[X] = n \times p$.

Variance of a Binomial Distribution

The measure of dispersion for a binomial distribution, given by the formula $\text{Var}(X) = n \times p \times (1 - p)$ (or $n \times p \times q$).

dbinom

An R function that computes the probability mass function (PMF), giving the exact probability of obtaining a specific number of successes ($x$) given trial size $n$ and probability $p$.

pbinom

An R function that calculates the cumulative probability mass function, giving the probability of observing less than or equal to a specific number of successes ($P(X \le x)$).

Bernoulli Distribution

A special case of the binomial distribution where the total number of trials is equal to one ($n = 1$).