Continuous Probability Distributions

Explanation

The continuous uniform distribution has constant probability density over the interval $[a, b]$. The probability density function is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$, and $0$ otherwise. The expected value is $E[X] = \frac{a+b}{2}$ and the variance is $\text{Var}(X) = \frac{(b-a)^2}{12}$. This distribution models situations where all values in an interval are equally likely, such as the position of a randomly thrown dart on a board, random arrival times within a time window, or measurement errors uniformly distributed within tolerances.