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: fX(x) = $ \lambda*e^{-\lambda*x} $, x >= 0 ; fX(x) = 0 , else
CDF: FX(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 $
