3) General Random Variables

3.1 Continuous Random Variables and PDFs

• Continuous Random Variable: A random variable $X$ is continuous if there exists a probability density function (PDF) $f_X(x)$ such that for any interval $(a,b)$: $$P(a<X<b)={\int }_a^b{f}_X(x)dx$$
  • $f_X(x)\ge 0$ and
  \int_{-\infty}^{\infty} f_X(x) , dx = 1 - The probability that $X$ takes a specific value is always zero: $P(X = a) = 0$.

3.2 Cumulative Distribution Functions (CDF)

• The Cumulative Distribution Function (CDF) of a random variable $X$ is defined as: $$F_X(x)=P(X\le x)$$
  • For discrete random variables: $$F_X(x)=\sum _{k\le x}P(X=k)$$
  • For continuous random variables: $$F_X(x)={\int }_{-\infty }^x{f}_X(t)dt$$

3.3 Normal Random Variables

• Normal (Gaussian) Random Variable: A continuous random variable $X$ is normal if it has a PDF: $$f_X(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$$
  • The parameters $\mu$ and ${\sigma }^2$ are the mean and variance, respectively.

3.4 Joint PDFs of Multiple Random Variables

• The Joint PDF of two continuous random variables $X$ and $Y$ is a function $f_{X,Y}(x,y)$ such that: $$P((X,Y)\in A)={\int }_A{f}_{X,Y}(x,y)dxdy$$
• Marginal PDFs can be found by integrating the joint PDF: $$f_X(x)={\int }_{-\infty }^{\infty }{f}_{X,Y}(x,y)dy$$

3.5 Conditioning

• The Conditional PDF of $X$ given $Y=y$ is: $$f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)},\text{if }{f}_Y(y)>0$$
  • Total Expectation: $$E[X]={\int }_{-\infty }^{\infty }E[X|Y=y]f_Y(y)dy$$

3.6 Continuous Bayes’ Rule

• Bayes’ Rule for Continuous Variables: $$f_{Y|X}(y|x)=\frac{f_{X|Y}(x|y)f_Y(y)}{f_X(x)}$$