Overview
This lecture explains cumulative distribution functions (CDFs), their difference from probability density functions (PDFs), and demonstrates how to use them for uniform and exponential distributions.
Cumulative Distribution Functions (CDF)
- The CDF calculates the area under a probability curve to the left of a specified point.
- For any continuous probability distribution, total area under the curve equals 1.
- The CDF gives the cumulative probability up to a value X, representing P(X ≤ x).
Probability Density Functions (PDF)
- The PDF, denoted f(x), describes the shape of the distribution.
- For a uniform distribution, f(x) = 1/(B - A) for A ≤ x ≤ B.
- For an exponential distribution, f(x) = λe^(−λx), where λ (lambda) is the rate parameter (λ = 1/mean).
Uniform Distribution
- The CDF for a uniform distribution is: P(X ≤ x) = (x - A)/(B - A) for A ≤ x ≤ B.
- The PDF for a uniform distribution is constant: f(x) = 1/(B - A).
Exponential Distribution
- The CDF for an exponential distribution is: P(X ≤ x) = 1 - e^(−λx).
- To find the probability X is between a and b: P(a < X < b) = CDF(b) - CDF(a) = [1 - e^(−λb)] - [1 - e^(−λa)].
- The area to the right of x is e^(−λx).
Properties of Continuous Distributions
- Probability that X equals any exact value is zero: P(X = a) = 0.
- Probability for intervals (e.g., P(a < X < b)) uses the difference between CDFs at b and a.
Key Terms & Definitions
- CDF (Cumulative Distribution Function) — Function giving the probability a random variable is less than or equal to a value.
- PDF (Probability Density Function) — Function showing the likelihood of a random variable taking a specific value; describes the distribution's shape.
- Uniform Distribution — A distribution where all intervals of equal length are equally probable.
- Exponential Distribution — A distribution often used for modeling time until an event, defined by a rate parameter λ.
- Rate Parameter (λ) — In exponential distribution, λ = 1/mean; governs the shape and scale.
Action Items / Next Steps
- Practice deriving and applying CDF and PDF formulas for uniform and exponential distributions.
- Review textbook sections on probability distributions, focusing on CDF and PDF distinctions.