11. Continuous Distributions
Definition
X∼U(a,b) means that X can take any value in the interval (a,b), with probability equally dense everywhere in the interval density function.
For example, take the following density function: f(x)={0b−a1b>x<aa<x<b

The cumulative distribution function is: F(x)=⎩⎨⎧0∫axb−a1dt=b−atax=b−ax−a1x<aa<x<bx>b
F(x)E[X]V(X)σ(X)=⎩⎨⎧0b−ax−a1x<aa<x<bx>b=2a+b=12(b−a)2=23b−a
Exponential Distribution
X∼Exp(λ)
\[\begin{aligned}f(x)&=\cases{\lambda e^{-\lambda x} &{$x>0$}\\0&$x\le0$}\\F(x)&=\cases{1-e^{-\lambda x} &{$x>0$}\\0&$x\le0$}\\E[X]&=\frac1\lambda\\V(X)&=\frac1{\lambda^2}\end{aligned}\]