Continuous Random Variables

Properties

Probability Density Function

• $f_X:\mathbb{R}\to \mathbb{R}$ if
  • $f_X(x)\ge 0$ for every $x\in \mathbb{R}$
  • ${\int }_{-\infty }^{\infty }{f}_X(x)=1$
  • $P(a\le x\le b)={\int }_a^b{f}_X(x)dx$ for all $a,b\in \mathbb{R}$
• Note: $P(X=x)=0$

Probability Distribution Function

• $F_X(b)=P(-\infty \le x\le b)={\int }_{-\infty }^b{f}_X(x)dx$ for every $b\in \mathbb{R}$

Continuous Expectation

• $\mathbb{E}[X]={\int }_{-\infty }^{\infty }x{f}_X(x)dx$
  • assuming that ${\int }_{-\infty }^{\infty }|x|f_X(x)dx<\infty$
• $\mathbb{E}[g(x)]={\int }_{-\infty }^{\infty }g(x)f_X(x)dx$
• $V(X)=\mathbb{E}[(X-\mathbb{E}X{)}^2]={\int }_{-\infty }^{\infty }(x-\mathbb{E}X{)}^2{f}_X(x)dx$

Distributions

Uniform

• All points equally likely to be chosen
• $X=\text{ points chosen in }(a,b)$
• $\mathcal{X}=(a,b)$

Probability:

$$\begin{array}{rl}P& =(X\in (c,d))=\frac{d-c}{b-a}\\ & \\ & \\ F_X(x)& =\{\begin{array}{ll}0& x<a\\ \frac{x-a}{b-a}& a\le x\le b\\ 1& b<x\end{array}\\ & \\ f_X(x)& =\frac{1}{b-a},a\le x\le b\end{array}$$

Expectation:

$$\begin{array}{rl}\mathbb{E}(X)& =\frac{a+b}{2}\\ V(X)& =\frac{(b-a{)}^2}{12}\end{array}$$

Exponential

• Probability decreases exponentially as interval increases
• Memoryless (like geometric)

Gamma

• Most values are concentrated around the mean

Normal

Beta

• good model for quantities that are proportions