Introduction to Random Variables and Discrete Probability Distributions

Vocabulary flashcards covering random variables, discrete versus continuous classifications, expected value, variance, and the binomial distribution along with its R implementation.

Random Variable

A mathematical concept that models uncertainty by assigning a numerical value to each outcome of a probability experiment.

Discrete Random Variable

A random variable that has a finite or countable number of possible outcomes that can be listed (e.g., counting birds, species biodiversity counts, or binary presence/absence indicators).

Continuous Random Variable

A random variable that has an uncountable number of possible outcomes, can take any value with infinite decimal points, and is typically represented by an interval on a number line (e.g., length, temperature, or earthquake magnitude).

Probability Distribution

A mathematical function (such as a PMF or PDF) that gives the probability of all possible values that a random variable can take across its sample space.

Principles of a Discrete Probability Distribution

The core rules stating that the probability of each possible outcome must be between $0$ and $1$, and the sum of the probabilities of all possible outcomes across the sample space must equal $1$.

Expected Value

The long-term average outcome or mean of a random variable, calculated as a weighted mean where each possible outcome is multiplied by its probability of occurrence: $E[X] = \sum x_i P(x_i)$.

Variance of a Discrete Random Variable

A measure of variability calculated by multiplying the squared deviation of each outcome from the mean by its respective probability and summing them up: $\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$.

Empirical Probability Distribution

A probability distribution constructed directly from observed data or frequency counts (e.g., counting birds in the field or tallying movie ratings).

Theoretical Probability Distribution

A probability distribution defined by a mathematical function with specific parameters that describe idealized scenarios (such as the binomial distribution).

Binomial Experiment

A probability scenario that meets four conditions: (1) each trial has only two outcomes (success or failure / $1$ or $0$), (2) it is repeated for a fixed number of trials ($n$), (3) trials are mutually independent, and (4) the probability of success ($p$) is identical for each trial.

Binomial Distribution

A discrete probability distribution that models the number of successful trials ($k$ or $X$) out of a fixed total number of independent trials ($n$).

Parameter $q$

The probability of failure on a single trial in a binomial distribution, defined mathematically as $q = 1 - p$.

Binomial Probability Mass Function (PMF)

The mathematical formula that gives the exact probability of observing $k$ successes in $n$ independent trials: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$.

Expected Value of a Binomial Distribution

The theoretical mean of a binomial distribution, calculated by multiplying the fixed number of trials by the probability of success: $E[X] = n \times p$.

Variance of a Binomial Distribution

The measure of spread for a binomial distribution, calculated using 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), yielding the exact probability of observing a specific number of successes ($x$) given the parameters size ($n$) and prob ($p$).

pbinom

An R function that calculates the cumulative probability distribution, 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 experiment consists of exactly one single trial ($n = 1$).