Contents
- 1 Cumulative Density Function (CDF)
- 2 Exponential RV
- 3 Gaussian RV
- 4 PDF Properties
- 5 Theorem of Total Probability for Continuous Random Variables
- 6 Conditioning a Random variable on an Event
- 7 Conditioning a Random variable on another Random variable
- 8 Shifting and Scaling of Random Variables
- 9 Addition of Discrete and Continuous Random Variables
- 10 Other Useful Things
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}} $
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 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, 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 \, $
- $ E[Y] = aE[X]+b \, $
- $ Var(X) = a^2 E[X^2] \, $
Addition of Discrete and Continuous 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 $
