Biostatistics: Continuous Probability Distributions

This set of vocabulary flashcards covers the fundamental concepts of continuous probability distributions, including uniform, exponential, and normal distributions, along with the empirical rule and Z-score calculations.

Continuous random variables

Variables that can take in any value in an interval such that all their possible values cannot be listed, typically used for measurements such as height, weight, and temperature.

Probability density function (PDF)

A smooth curve that represents the probability distribution of a continuous random variable, where the area under the curve represents probabilities and the total area equals $1$.

Uniform Probability Distribution

A continuous distribution where all outcomes are equally likely and the probability is proportional to the interval's length, resulting in a rectangular density curve.

Expected value of X (Uniform Distribution)

The average value of a uniform distribution calculated as $E(x) = \frac{a+b}{2}$ where $a$ and $b$ are the endpoints.

Variance of X (Uniform Distribution)

The measure of spread for a uniform distribution calculated as $\frac{(b-a)^2}{12}$.

Exponential Distribution

A distribution often used to describe the time or distance until some event happens, with the PDF defined as $f(x; \text{λ}) = \text{λ}e^{-\text{λ}x}$ for $x ≥ 0$.

Normal Probability Distribution

Also called the Gaussian probability distribution after Karl Friedrich Gaus, it is a symmetric, bell-shaped distribution defined by its mean ($μ$) and standard deviation ($σ$).

Mean ($μ$)

The parameter that determines the center and the highest point of a normal distribution curve; it is also equivalent to the median and mode.

Standard deviation ($σ$)

The parameter that determines the spread or width of a normal distribution, where larger values result in wider, flatter curves.

Inflection points

The points on a normal curve at which the slope changes direction, occurring at $μ - σ$ and $μ + σ$.

Empirical Rule (68.26%)

The characteristic stating that approximately $68.26\%$ of values in a normal distribution fall within $±1$ standard deviation ($μ ± 1σ$) of the mean.

Empirical Rule (95.44%)

The characteristic stating that approximately $95.44\%$ of values in a normal distribution fall within $±2$ standard deviations ($μ ± 2σ$) of the mean.

Empirical Rule (99.72%)

The characteristic stating that approximately $99.72\%$ of values in a normal distribution fall within $±3$ standard deviations ($μ ± 3σ$) of the mean.

Standard Normal Probability Distribution

A specific normal distribution that has a mean of $0$ and a standard deviation of $1$, designated by the letter $z$.

Z score

A standardized value representing the number of standard deviations an observation $x$ is from the mean, calculated as $Z = \frac{x-μ}{σ}$.

NORMS.DIST

An Excel function used to compute the cumulative probability awarded to a specific $z$ value in a standard normal distribution.

NORMS.INV

An Excel function used to compute the $z$ value corresponding to a given cumulative probability in a standard normal distribution.

P(x = x) for Continuous Variables

In continuous probability distributions, the probability of a random variable taking an exact single value is always equal to $0$.