Cheat2 ECE302Fall2008sanghavi - Rhea

Cumulative Density Function (CDF)

• FX(x) = P(X <= x) = integral(-inf to inf) fX(y) dy
• 1 - FX(x) = P(X > x)
• lim_{x$ -> -\infty $} f_X(x) = 0 , lim_{x-> inf} f_X(x) = 0

Exponential RV

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 $

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  $