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Cumulative Density Function (CDF)
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==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)
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Exponential RV
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==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>
  
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==Gaussian RV==
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*The sum of many, small independent things
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*Parameters:
  
PDF Properties
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<math>E[X]=\mu</math>
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<math>Var[X]=\sigma^2</math>
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<math>f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}</math>
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==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  $

Alumni Liaison

Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal