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CDFs and PDFs in Distributions

Sep 1, 2025

Overview

This lecture explains cumulative distribution functions (CDFs) and their use in finding probabilities for continuous probability distributions, specifically comparing CDFs with probability density functions (PDFs), and applies these concepts to uniform and exponential distributions.

Cumulative Distribution Functions (CDF)

  • The CDF gives the total probability (area under the curve) to the left of a specific value X.
  • For continuous distributions, probability equals the area under the curve and totals 1.
  • The CDF accumulates probability from the left up to point X.

Probability Density Functions (PDF)

  • The PDF (f(x)) shows the shape of a probability distribution and its height at any given X.
  • For a continuous variable, probability at a single point is always 0; you need a range to get a nonzero probability.

Uniform Distribution

  • PDF for uniform distribution: f(x) = 1/(B-A) for A ≤ x ≤ B.
  • CDF for uniform distribution: P(X ≤ x) = (x - A)/(B - A).
  • The CDF represents the area of a rectangle with base (x-A) and height f(x).

Exponential Distribution

  • Exponential PDF: f(x) = λe^(–λx), where λ (lambda) is the rate parameter (λ = 1/mean).
  • Exponential CDF: P(X ≤ x) = 1 – e^(–λx).
  • Area to the right of X: e^(–λx).
  • Probability between A and B: P(A < X < B) = CDF at B minus CDF at A = [1 – e^(–λB)] – [1 – e^(–λA)].

Key Properties and Additional Notes

  • For continuous distributions, P(X = a) = 0; only intervals yield nonzero probability.
  • P(a ≤ X ≤ b) is equivalent to P(a < X < b) in continuous cases, since single points have zero probability.

Key Terms & Definitions

  • Probability Density Function (PDF) — describes the shape and height of the probability distribution.
  • Cumulative Distribution Function (CDF) — gives total probability to the left of a value X, representing accumulated area.
  • Rate Parameter (λ) — parameter for exponential distributions, λ = 1/mean.

Action Items / Next Steps

  • Practice calculating PDF and CDF for uniform and exponential distributions.
  • Review how to find probability in intervals using CDFs.

Full transcript