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: f_X(x) = $ \lambda*e^{-\lambda*x} $, x >= 0 ; f_X(x) = 0 , else

CDF: F_X(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 $

Back to ECE302 Fall 2008 Prof. Sanghavi