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📚 Mostly Harmless Statistics
Chapters ▾
1 Introduction to Data2 Organizing Data3 Descriptive Statistics4 Probability5 Discrete Probability Distributions6 Continuous Probability Distributions7 Confidence Intervals for One Population8 Hypothesis Tests for One Population9 Hypothesis Tests and Confidence Intervals for Two Populations10 Chi-Square Tests11 Analysis of Variance12 Correlation and Regression13 Nonparametric Tests

Chapter 6: Continuous Probability Distributions

A continuous random variable (usually denoted as X) is a variable that has an infinite number of random values in an interval of numbers. There are many different types of continuous distributions. To be a valid continuous distribution the total area under the curve has to be equal to one and the function’s y-values need to be positive.

For example, we may have a random variable that is uniformly distributed so we could use the Uniform distribution that looks like a rectangle. See Figure 6-1.

Uniform distribution graph: a horizontal blue segment at height 1/(b-a) spans between a and b, with dashed vertical lines dropping to open circles at a and b on the x-axis, outlining a rectangle.

Figure 6-1

We may want to model the time it takes customer service to complete a call with the exponential distribution. See Figure 6-2.

Three exponential distribution curves for mu = 4 in blue, mu = 1.2 in green, and mu = 0.5 in magenta over x from 0 to 10; the mu = 0.5 curve starts highest near 2 on the y-axis and drops fastest, while the mu = 4 curve starts lowest and decays slowest toward zero.

Figure 6-2

We may have standardized test scores that follow a bell-shaped curve like the Gaussian (Normal) Distribution. See Figure 6-3.

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University), hosted on LibreTexts (stats.libretexts.org) and licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.