Continuous distributions

A continuous uniform distribution spreads probability evenly over an interval \([a,b]\). Its density function is constant: \(f(x)=\frac{1}{b-a}\) for \(a\le x\le b\) and zero elsewhere. Because the total area under the curve must equal 1, the height of the rectangle is the reciprocal of the interval length. Probabilities are simply the proportion of the interval covered: \(P(c\le X\le d)=\frac{d-c}{b-a}\) when \(a\le c<d\le b\). The mean (expected value) is the midpoint \(\mu=\frac{a+b}{2}\) and the standard deviation is \(\sigma=\frac{b-a}{\sqrt{12}}\). Knowing these formulas lets you answer questions about “less than”, “between”, or “greater than” values without integrating.