Random Variable cdf

# Random Variable cdf - Cumulative Distribution Function of a Random Variable Definition FX x = P X x = P[cfw_s X s x Other terms are cdf and

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Cumulative Distribution Function of a Random Variable Definition: () [ ] [{ : ( ) }] X F x PX x P s Xs x = = −∞≤ Other terms are cdf and Probability Distribution Function (PDF).

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Example – Uniform Distribution Throw dart at a spinning wheel. X is the phase where the dart hits the wheel. Find the cdf of X . First we note the range of the random variable X, {0 2 } X Sx π = ≤≤ . Within the range, [ / 2 / 2] / 2 for any phase . Px X x x −∆ ≤ +∆ =∆ Therefore, [ ] 12 () P 0 2 2 00 X x x Fx X x x x = = < Homework: Plot the cdf of (1,3). U
Example - Bernoulli Let X be a Bernoulli random variable. 1 with prob 0 for a head p X for a tail = , 11 ( ) P[ ] 1 0 1 00 X x Fx X x p x x = = ≤< < Discrete RVs have discontinuities in their cdf’s. The values of their cdf’s are taken approaching from the right. 1 1 1 p

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Properties: 0 () 1 X Fx ≤≤ ()1 X F ∞= ( )0 X F −∞ = () X is a non-decreasing function of x X is continuous from the right: that is, 0 ( ) lim ( ) XX h Fb h =
probability density function ( pdf) Definition: 0 ( ) () () l im XX Fx d f x dx ∆→ +∆ − = Differentiate the cdf to get the pdf A pdf is the rate at which its cdf increases

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## This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.

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Random Variable cdf - Cumulative Distribution Function of a Random Variable Definition FX x = P X x = P[cfw_s X s x Other terms are cdf and

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