PDF and CDF: Normal and Uniform Distributions
F(x) = P(X ≤ x): the cumulative probability up to and including x.
Definition
F(x) = P(X ≤ x) is the area under the PDF f(t) from −∞ to x.
Equivalently: F(x) is the probability that a randomly drawn observation is at most x.
Setup
Compute
Height of the PDF
f(x) = 1b − a for a ≤ x ≤ b, and 0 otherwise
CDF
F(x) = x − ab − a for a ≤ x ≤ b
F(x) = 0 below a, and F(x) = 1 above b.
Mean
μ = a + b2
Standard deviation
σ = (b − a)²12
Probability of an interval = width × height
P(x₁ ≤ X ≤ x₂) = x₂ − x₁b − a = (x₂ − x₁) × 1b − a
Keep x₁ and x₂ inside [a, b]. Any part of an interval outside a to b has no area. Because X is continuous, P(X = x) = 0, so < and ≤ give the same answer.
Managerial implication
A note on letters: in U(a, b), the letters a and b are the distribution’s own endpoints (here 15 and 30 minutes). So the endpoints of the interval you are finding a probability for are called x₁ and x₂.
Pick a probability, then move the sliders
Change the distribution’s numbers
Practice
Select “New question” to begin.
The Empirical Rule
For any normal distribution X ~ N(μ, σ).
68.26%
within μ ± 1σ
one-sigma interval
95.44%
within μ ± 2σ
two-sigma interval
99.73%
within μ ± 3σ
three-sigma interval
The percentages match the slide, which doubles z-table areas (2 × 0.3413 and 2 × 0.4772). Computed exactly, they are 68.27% and 95.45%, which is what the sliders at the top of this page show. The difference is rounding only.
The Standard Normal Distribution and z-Scores
Any normal X ~ N(μ, σ) can be rescaled to Z ~ N(0, 1).
Standard normal distribution
Z ~ N(0, 1) is the normal distribution with mean 0 and standard deviation 1.
A z-score counts how many standard deviations a value x lies from the mean: positive above the mean, negative below it.
z = x − μσ
Going back from z to x: x = μ + z σ
Quick check
A few questions on the empirical rule (answer in percent, like 34.13) and on z-scores (answer to 2 decimals).
Select “New question” to begin.