(Conditioning a Random variable on an Event)
(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>
 +
* For Continuos Random Variable:
 +
  P(X > x) = <math> \int\limits_{x}^{\infty}f_X(x)dx </math>
 +
  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==

Revision as of 17:51, 22 October 2008

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 $

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  $
  • For Continuos 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 \, $

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, andm therefore,

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

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 \, $



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

Finding PDFs and CDFs of functions of Random Variables

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 $

Alumni Liaison

Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010