(Cumulative Density Function (CDF))
 
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[[Category:ECE302Fall2008_ProfSanghavi]]
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[[Category:probabilities]]
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[[Category:ECE302]]
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[[Category:cheat sheet]]
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=[[ECE302]] Cheat Sheet number 2=
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==Cumulative Density Function (CDF)==
 
==Cumulative Density Function (CDF)==
 
* <math>F_X(x) = P[X \leq x] = \int_{-\infty}^{\infty} f_x(t)dt</math>
 
* <math>F_X(x) = P[X \leq x] = \int_{-\infty}^{\infty} f_x(t)dt</math>
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<math>\lim_{x\rightarrow\infty}F_X(x) = 1 </math>
 
<math>\lim_{x\rightarrow\infty}F_X(x) = 1 </math>
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'''If X is discrete PX(k)''' = P(X<= k)-P(X<=k-1)
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                      = FX(k)-FX(k-1)
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'''Converting from CDF -> PDF :'''
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fX(x) = d FX(x)/dt  i.e. Take derivative of the CDF to get PDF
  
 
==Exponential RV==
 
==Exponential RV==
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*The first occurance of a very rare event, when trials happen very fast.
  
 
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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<math>f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}</math>
 
<math>f_X(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}</math>
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For 2 independent Gaussians:
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<math>Z=X+Y\!</math><br>
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<math>E[Z]=\mu_X +\mu_Y\!</math><br>
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<math>Var(Z)=\sigma^2_X+\sigma^2_Y</math>
  
 
==PDF Properties==
 
==PDF Properties==
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* For any subset B of the real line,  
 
* For any subset B of the real line,  
 
   <math> P(X\in B) = \int\limits_Bf_X(x)dx </math>
 
   <math> P(X\in B) = \int\limits_Bf_X(x)dx </math>
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* For Continuous Random Variable:
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  P(X > x) = <math> \int\limits_{x}^{\infty}f_X(x)dx </math>
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  P(X <= x) = <math> \int\limits_{-\infty}^{x}f_X(x)dx </math>
  
 
==Theorem of Total Probability for Continuous Random Variables==
 
==Theorem of Total Probability for Continuous Random Variables==
*<math>f_Y(y) = f_{Y|A}(y)P(A) + f_{Y|B}(y)P(B)\,</math>
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*<math>f_Y(y) = f_{Y|A}(y)P(A) + f_{Y|B}(y)P(B)\,</math>
 
*<math>f_X(x) = \int^\infty_{-\infty}f_{XY}(x,y)dy = \int^\infty_{-\infty}f_{X|Y}(x|y)f_Y(y)dy \,</math>
 
*<math>f_X(x) = \int^\infty_{-\infty}f_{XY}(x,y)dy = \int^\infty_{-\infty}f_{X|Y}(x|y)f_Y(y)dy \,</math>
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*<math>f_Y(y) = f_{Y|A1}(y)P(A1) + f_{Y|A2}(y)P(A2) + f_{Y|A3}(y)P(A3)+ ... + f_{Y|Ai}(y)P(Ai)\,</math>  if A1, A2, A3,...  is disjoint
  
 
==Conditioning a Random variable on an Event==
 
==Conditioning a Random variable on an Event==
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<math>f_{X|Y}(x)=P(X=x|A)=\frac {P({X=x}\cap A)}{P(A)}</math>
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The events <math>{X=x}\cap A </math> are disjoint for different values of x, their union is A, and,therefore,
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<math> P(A)=\sum_xP({X=x}\cap A)</math>
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<math> \sum_xP_{x|A}(x)=1 </math>
  
 
==Conditioning a Random variable on another Random variable==
 
==Conditioning a Random variable on another Random variable==
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<math>f_{X|Y}(x|y)=\dfrac {f_{XY}(x,y)}{f_{Y}(y)}</math>
  
 
==Shifting and Scaling of Random Variables==
 
==Shifting and Scaling of Random Variables==
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<math>f_Y(y)=\dfrac{1}{|a|}f_Y(\dfrac{y-b}{a})\!</math>
  
 
*<math>E[Y] = aE[X]+b \,</math>
 
*<math>E[Y] = aE[X]+b \,</math>
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*<math>Var(X) = a^2 E[X^2] \,</math>
 
*<math>Var(X) = a^2 E[X^2] \,</math>
  
==Finding PDFs and CDFs of functions of Random Variables==
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== Addition of Continuous Random Variables ==
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If  X and Y are '''continuous''' and independent random variables and Z  = X + Y then
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*<math>f_Z(z) = \int^\infty_{-\infty} f_X(x)f_Y(z-x) dx</math>
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== Addition of Discrete Random Variables ==
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If  X and Y are '''discrete''' and independent random variables and Z  = X + Y then
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'''<math>f_Z(z) = \sum_X f_X(x)f_Y(z-x)</math>'''
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== Continuous Bayes' rule: ==
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'''<math>f_{X|Y}(x|y)=((f_X(x)).f_{Y|X}(y|x))/f_Y(y)</math> '''
  
 
==Other Useful Things==
 
==Other Useful Things==
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*<math>f_Y(y) = \int^\infty_{-\infty} f_{XY}(x,y) dx</math>
 
*<math>f_Y(y) = \int^\infty_{-\infty} f_{XY}(x,y) dx</math>
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*<math>E(g(x))=\int^\infty_{-\infty} g(x)f_X(x,y) dy</math>
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----
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[[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]]

Latest revision as of 13:05, 22 November 2011


ECE302 Cheat Sheet number 2

Cumulative Density Function (CDF)

  • $ F_X(x) = P[X \leq 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 $

If X is discrete PX(k) = P(X<= k)-P(X<=k-1)

                      = FX(k)-FX(k-1)

Converting from CDF -> PDF :

fX(x) = d FX(x)/dt i.e. Take derivative of the CDF to get PDF

Exponential RV

  • The first occurance of a very rare event, when trials happen very fast.

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}} $

For 2 independent Gaussians:

$ Z=X+Y\! $
$ E[Z]=\mu_X +\mu_Y\! $
$ Var(Z)=\sigma^2_X+\sigma^2_Y $

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  $
  • For Continuous Random Variable:
 P(X > x) = $  \int\limits_{x}^{\infty}f_X(x)dx  $
 P(X <= x) = $  \int\limits_{-\infty}^{x}f_X(x)dx  $

Theorem of Total Probability for Continuous Random Variables

  • $ f_Y(y) = f_{Y|A}(y)P(A) + f_{Y|B}(y)P(B)\, $
  • $ f_X(x) = \int^\infty_{-\infty}f_{XY}(x,y)dy = \int^\infty_{-\infty}f_{X|Y}(x|y)f_Y(y)dy \, $


  • $ f_Y(y) = f_{Y|A1}(y)P(A1) + f_{Y|A2}(y)P(A2) + f_{Y|A3}(y)P(A3)+ ... + f_{Y|Ai}(y)P(Ai)\, $ if A1, A2, A3,... is disjoint

Conditioning a Random variable on an Event

$ f_{X|Y}(x)=P(X=x|A)=\frac {P({X=x}\cap A)}{P(A)} $

The events $ {X=x}\cap A $ are disjoint for different values of x, their union is A, and,therefore,

$ P(A)=\sum_xP({X=x}\cap A) $

$ \sum_xP_{x|A}(x)=1 $

Conditioning a Random variable on another Random variable

$ f_{X|Y}(x|y)=\dfrac {f_{XY}(x,y)}{f_{Y}(y)} $

Shifting and Scaling of Random Variables

Let $ Y=aX+b \, $


$ f_Y(y)=\dfrac{1}{|a|}f_Y(\dfrac{y-b}{a})\! $

  • $ E[Y] = aE[X]+b \, $
  • $ Var(X) = a^2 E[X^2] \, $

Addition of Continuous Random Variables

If X and Y are continuous and independent random variables and Z = X + Y then


  • $ f_Z(z) = \int^\infty_{-\infty} f_X(x)f_Y(z-x) dx $

Addition of Discrete Random Variables

If X and Y are discrete and independent random variables and Z = X + Y then

$ f_Z(z) = \sum_X f_X(x)f_Y(z-x) $

Continuous Bayes' rule:

$ f_{X|Y}(x|y)=((f_X(x)).f_{Y|X}(y|x))/f_Y(y) $

Other Useful Things

If X and Y are indepdent of each other, then

  • $ E[XY] = E[X]E[Y]\! $
  • $ E[X] = \int^\infty_{-\infty}x*f_X(x)dx\! $
  • $ Var(X) = E[X^2] - (E[X])^2\! $

Marginal Probability Density Functions:

  • $ f_X(x) = \int^\infty_{-\infty} f_{XY}(x,y) dy $
  • $ f_Y(y) = \int^\infty_{-\infty} f_{XY}(x,y) dx $
  • $ E(g(x))=\int^\infty_{-\infty} g(x)f_X(x,y) dy $

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