Cumulative Density Function (CDF)

• $ F_X(x) = P[X <= x] = \int_{-\infty}^{\infty} f_x(t)dt $
• $ 1 - F_X(x) = P[X > x] $

$ \lim_{x\rightarrow-\infty}f_X(x) = 0 $

$ \lim_{x\rightarrow\infty}f_X(x) = 1 $

Exponential RV

PDF: f_X(x) = $ \lambda*e^{-\lambda*x} $, x >= 0 ; f_X(x) = 0 , else

CDF: F_X(x) = $ 1-e^{-\lambda*x} $

• E[X] = 1/$ \lambda $ , var(X) = 1/($ \lambda)^2 $

Gaussian RV

• The sum of many, small independent things
• Parameters:

$ E[X]=\mu $ $ Var[X]=\sigma^2 $

$ f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}} $

PDF Properties

• $ f_X(x)\geq 0 $ for all x
• $ \int\limits_{-\infty}^{\infty}f_X(x)dx = 1 $
• If $ \delta $ is very small, then

 $  P([x,x+\delta]) \approx f_X(x)\cdot\delta $

• For any subset B of the real line,

 $  P(X\in B) = \int\limits_Bf_X(x)dx  $

Theorem of Total Probability for Continuous Random Variables

Conditioning a Random variable on an Event

Conditioning a Random variable on another Random variable

Shifting and Scaling of Random Variables

Finding PDFs and CDFs of functions of Random Variables=

Other Useful Things

• $ E[X] = \int\limitsABx*f_X(x)dx $