Uniform distribution

• How to Find Probabilities for a Uniform Distribution
  For a continuous uniform distribution on [a, b], find an event’s probability by dividing its overlap length with [a, b] by b − a.
• How to Find a Confidence Interval for a Uniform Distribution
  For independent Uniform(0, θ) data, use the sample maximum to derive an exact confidence interval, with formulas and a worked 95% example.
• What Is the Entropy of the Sum of Two Uniform Random Variables?
  For independent X and Y uniform on [0,1], their sum has a triangular density and differential entropy 1/2 nat (about 0.721 bits).
• Why Can’t I Compute the Expected Value from a Uniform MGF?
  For X uniform on [a,b], the MGF has an apparent 0/0 at t=0; extend it by continuity, then differentiate to get E[X]=(a+b)/2.
• What Is the MLE for a Uniform(θ, 0) Distribution?
  For iid data from Uniform(θ, 0) with θ < 0, the likelihood is maximized at the sample minimum; if endpoints are reversed, it is the maximum.
• How to Estimate Uniform Distribution Parameters by Moments
  For a uniform model, match sample moments to population moments; see formulas for U(0, θ) and U(a, b), with a worked example.
• What Is the Minimal Sufficient Statistic for Uniform(θ, θ)?
  For Uniform(θ, θ), the endpoints coincide, so the model is degenerate; for a nonzero interval, the minimal statistic depends on its endpoints.
• How to Find the MLE of a Uniform Distribution
  For an i.i.d. sample from Uniform(0, θ), the maximum-likelihood estimate is the sample maximum; endpoint estimates differ when both bounds are unknown.
• What Is the MGF of a Uniform Distribution?
  For a continuous uniform random variable on [a,b], the moment-generating function is (e^{bt}-e^{at})/[t(b-a)], with value 1 at t=0.