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| − | Cumulative Density Function (CDF) | + | ==Cumulative Density Function (CDF)== |
* 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) | ||
| Line 7: | Line 7: | ||
| − | Exponential RV | + | ==Exponential RV== |
PDF: f<sub>X</sub>(x) = <math>\lambda*e^{-\lambda*x}</math>, x >= 0 ; f<sub>X</sub>(x) = 0 , else | PDF: f<sub>X</sub>(x) = <math>\lambda*e^{-\lambda*x}</math>, x >= 0 ; f<sub>X</sub>(x) = 0 , else | ||
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* E[X] = 1/<math>\lambda</math> , var(X) = 1/(<math>\lambda)^2</math> | * E[X] = 1/<math>\lambda</math> , var(X) = 1/(<math>\lambda)^2</math> | ||
| + | ==Gaussian RV== | ||
| + | *The sum of many, small independent things | ||
| + | *Parameters: | ||
| − | PDF Properties | + | <math>E[X]=\mu</math> |
| + | <math>Var[X]=\sigma^2</math> | ||
| + | |||
| + | <math>f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}</math> | ||
| + | |||
| + | |||
| + | ==PDF Properties== | ||
* <math> f_X(x)\geq 0 </math> for all x | * <math> f_X(x)\geq 0 </math> for all x | ||
* <math> \int\limits_{-\infty}^{\infty}f_X(x)dx = 1</math> | * <math> \int\limits_{-\infty}^{\infty}f_X(x)dx = 1</math> | ||
Revision as of 18:01, 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 $
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 $
