| Line 2: | Line 2: | ||
* FX(x) = P(X <= x) = integral(-inf to inf) fX(y) dy | * FX(x) = P(X <= x) = integral(-inf to inf) fX(y) dy | ||
* 1 - FX(x) = P(X > x) | * 1 - FX(x) = P(X > x) | ||
| − | + | ||
| + | lim<sub>x-> -inf</sub> F<sub>X</sub>(x) = 0 | ||
| + | lim<sub>x-> inf</sub> F<sub>X</sub>(x) = 1 | ||
Revision as of 10:51, 21 October 2008
Cumulative Density Function (CDF)
- FX(x) = P(X <= x) = integral(-inf to inf) fX(y) dy
- 1 - FX(x) = P(X > x)
limx-> -inf FX(x) = 0 limx-> inf FX(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 $
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 $
