AP Statistics: Random Variables, Density Curves, and Normal Distributions

Vocabulary practice flashcards covering discrete and continuous random variables, expected value, standard deviation, density curves, uniform distributions, the Empirical Rule, and normal distribution calculations.

Discrete Random Variable

A random variable that takes a countable number of distinct values with gaps between them.

Discrete Probability Distribution

A model that lists all possible values of a discrete random variable along with their corresponding probabilities, where all individual probabilities are between $0$ and $1$, and their sum is equal to $1$.

Expected Value of a Discrete Random Variable ($\mu_X$)

The mean or long-run average outcome of a random variable after repeating a chance process many times, calculated as $\mu_X = \sum x_i \cdot P(x_i)$.

Standard Deviation of a Discrete Random Variable ($\sigma_X$)

A measure of variability representing the typical distance that outcomes of a discrete random variable vary from the mean $\mu_X$, calculated as $\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}$.

Continuous Random Variable

A random variable that can take on any value across a continuous interval, having an infinite number of possible values with no gaps between them.

Uniform Distribution

A continuous probability distribution where every outcome across a specific interval has an equal likelihood, resulting in a constant height and a rectangular density curve.

Probability Density Curve

A mathematical curve that models the distribution of a continuous random variable, where the total area underneath the curve is exactly equal to $1$ (or $100\%$) and the area over any specific interval represents the probability of that interval.

Percentile in a Continuous Distribution

A specific value on the horizontal axis of a probability density curve such that a designated percentage of the total area under the curve lies at or below that value.

Mean vs. Median in a Symmetric Distribution

The relationship in a bell-shaped or balanced symmetric distribution where the center of balance and the equal-area point align, resulting in $\text{mean} \approx \text{median}$.

Mean vs. Median in a Skewed Right Distribution

The relationship in a right-skewed distribution where unusually large values in the long right tail pull the center of balance upward, making $\text{mean} > \text{median}$.

Mean vs. Median in a Skewed Left Distribution

The relationship in a left-skewed distribution where unusually small values in the long left tail pull the center of balance downward, making $\text{mean} < \text{median}$.

Normal Distribution

A continuous, bell-shaped, and symmetric density curve specified entirely by its mean ($\mu$) and standard deviation ($\sigma$), denoted as $N(\mu, \sigma)$.

Empirical Rule (68-95-99.7 Rule)

A property of approximately normal distributions stating that roughly $68\%$ of observations lie within $1$ standard deviation of the mean ($\mu \pm 1\sigma$), roughly $95\%$ lie within $2$ standard deviations ($\mu \pm 2\sigma$), and roughly $99.7\%$ lie within $3$ standard deviations ($\mu \pm 3\sigma$).

Standard Normal Distribution

A special normal distribution where the mean is $0$ and the standard deviation is $1$, denoted as $N(0, 1)$.

$z$-score (Standardized Score)

A standardized value indicating how many standard deviations a raw data value $x$ falls above or below the population mean $\mu$, calculated using the formula $z = \frac{x - \mu}{\sigma}$.

Inverse Normal Calculation (Boundary Value)

The process of identifying a cutoff value $x$ given a specified area or percentile under a normal curve by first identifying the corresponding $z$-score and then solving for $x$ in the formula $z = \frac{x - \mu}{\sigma}$.