Continuous Random Variables

Let x ~ U(0, 0.5), PDF =

Density is a function that is a measure of likelihood, NOT probability

• Area is probability

• Density is the height

E[x] = integral of x * f(x)dx

var(x) = integral of (x-mu)² * f(x)dx

Expectation and Dependence

• Expectation: The expected (probability-weighted average) value

• Operator: mapping between spaces

Covariance: Measures the joint variability of two random variables $Cov\left(X,Y\right)=E\left\lbrack\left(X-\mu_{x}\right)\left(\left.Y-\mu_{x}\right)\right\rbrack=E\left\lbrack XY\right\rbrack-E\left\lbrack X\right\rbrack E\left\lbrack Y\right\rbrack\right.$

If Cov(X,Y) > 0, move together, if < 0 move opposite, if 0 uncorrelated

Independence: When X is perpendicular to Y $\displaylines{E\left\lbrack XY\right\rbrack=E\left\lbrack X\right\rbrack E\left\lbrack Y\right\rbrack\\ }$

Standard Normal Distribution - Gauss (Law of Errors)

• Mean = 0

• Variance / Standard Deviation = 1

• $f\left(x\right)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{\left(x-\mu\right)^2}{2\sigma^2}}$

• If you know the CDF, you can recover the PDF for the normal distribution

Joint CDF

CDF: Mapping between cutoffs and percentages

Joint PMF: Probability that two different outcomes occur together

Joint Distribution

• We have two continuous variables X and Y, and we know that the joint density function is f(x,y)=x+y. Is x independent from y?

  • If x and y are independent, the joint PMF is the outer product of the marginals

• X and Y are measured on the same outcome

• The joint CDF always exists

• $\displaylines{F_{X,Y}\left(x,y\right)=P\left(X\le x,\right.Y\le y)}$

Joint Normal

• (X,Y) is jointly normal if every linear combination aX + bY has a normal distribution

Conditional Density

• $f_{x\left|y\right.}\left(x\left|y\right.\right)\frac{f_{x,y}\left(x,y\right)}{f_{y}y}$

Conditional Expectation

• $E\left\lbrack X\left|Y=y\right.\right.]=\int x\cdot f\left(x\left|y\right.\right)$

IID = Independent and identically distributed