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# ELEG310 Week 4 notes ELEG310

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This 5 page Class Notes was uploaded by Cindy Nguyen on Sunday March 6, 2016. The Class Notes belongs to ELEG310 at University of Delaware taught by Dr. Daniel Weile in Spring 2016. Since its upload, it has received 30 views. For similar materials see Random Signals and Noise in Electrical Engineering at University of Delaware.

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Date Created: 03/06/16

ELEG 310 Week 4 Lectures 7 and 8 Review the Probability Mass Function from Chapter 3. New Topic: Conditional Probability Mass function Let C be the event that can happen such that C ∋ P[C] > 0. ???? = x is an atomic event. P???? [????|C] = P[???? = x|C] ={????=????}|????] P ????[????]≥0 ????[????] ∑ P????[????|C] = ∑ ????[{????=????}|????] ????∈???????? ????∈???? ????[????] = 1 ∑ ????[{???? = ????}] = 1 ????[????] ????∈???? ????[{????=????}|????] P[???? = B|C] = ????∈???? ????[????] = ∑ ????∈???? P ????[????|C] where B ⊂ ???? ???? (See textbook example 3.25 Device Lifetimes) Random Variable Review – mapping Uses semi-infinite intervals ????????(????) = P[????≤ x] notation for cdf, cumulative distribution function. Some constant Probability of outcome Used if the underlying space is discrete, continuous, or a combination of the two. It is better to use the pmf, or probability mass function. If we have a discrete variable, ???? (????) = P[????≤ x] ???? = ∑ ????????∈???? P ????[????k] = ∑ ???? P????[????k] u[x –kx ] k is a discrete When we reach a Unit step outcome possible function outcome, x will 0 ???? < 0 suddenly jump U(x) { 1 ???? ≥ 0 up If we have a continuous variable, it’s written as integral: ???? ????????(????) = ∫−∞ ???? ???? ???????? ???????????? is a new function known as the “probability density function,” or pdf. It is the density of probability in a particular place. The pdf, probability density function, is the derivative of the cdf, cumulative density function. P[x ???? ≤ x+h] ????????(x+h) - ???? (????) = ????????(x+h) − ???? (h) The probability ℎ that???? is in a small interval in the vicinity of x Properties: (i) The probability density ???? ???? ≥ ????n ∀ ???? ???????????????????????? ???????????????????????????????? ???? ???? (ii) P [a < x ≤∫b]???????????? ???????? ???? ???? (iii) P [???? < x????????(????) = ∫ ???????????? ???????? ∞ ???? (iv) ∫ ???????????? ???????? = 1 −∞ Examples of Random Variable and Probability Density Uniform Random Variable 1 ( ) ???? ≤ ???? ≤ ???? ???????????? = { ????−???? 0 ????????ℎ???????????????????????? If x < a, then the cdf????is ???? (???????? = ???? ???? ???????? If ???? ≤ ???? ≤ ????, then ???? (????) =∫???? ???? ???? ???????? = ∫ ???? 1 dt = ????−???? ???? −∞ ???? 0 ????−???? ????−???? If If x > ????, ???? ???? = 1 0 ???? < ???? ????−???? So, ???????????? = { ????−???? ???? ≤ ???? ≤ ???? 1 ???? > ???? Exponential Random Variable If P[????] = ???????????? x>0 Then ????[< ] = 1 − ???????????? x>0 So the cdf of this exponential random variable is −???????? ???? ???? = { 1 − ???? ???? > 0 ???? 0 ????????ℎ???????????????????????? Otherwise, Pdf differentiate ???????????? = ???? ′(????) = { 0 ???? < 0 ???????? −???????? ???? ≥ 0 Laplacian Random Variable We want a 2-sided exponential. −????|????| ???????????? = C???? where C is some constant e.g. Find C, to make sure pdf integrates to 1. ∞ 0 ???????? ∞ −???????? 1 =∫−∞ ???????????? ???????? = −∞ C???? ???????? + ∫0 C???? dx ???????????? 0 ????−???????? ∞ =[???? ???? ]−∞ + [????−???? ] 0 1 1 = C [ − ] ???? −???? 2 1 = C ???????? ???? So C = and ???? ???? = ????−????|????| 2 2 Now to find cdf: integrate pdf ???? (????) = ∫???? ???? ???? ???????? ???? ∞ ???? For x<0 ???? ???? ???????? 1 ???????? ????????(????) = ∫∞ ???? ???????? = ???? 2 2 For x=0 ???? (0) = 1 ???? 2 For x>0 ???????????? = ???? ????????????????) ∞ ???? = ∫−∞ ???????????? ???????? + ∫0 ???? ???? ???????? 1 ???? ???? −???????? = 2+ 2∫0 ???? ???????? −???????? = + [???? ???? ]???? 2 2 −???? 0 1 1 ????−???????? = + − 2 2 −????????2 = 1 − ???? 2 1 ???????????? ???? = 0 So, ???? (????) = { 2 ???? ????−???????? 1 − 2 ???? < 0 Pdf if more intuitive than cdf. Pdf of discrete random variable is hard because it’s discontinuous. The pdf for a discrete random variable is defined by ???????????? =) ∑???? P ????[k ] u[xk– x ] ???? ???????????? ) =???????? ???????????? = ∑ ???? P???? [k ] ????[xk– x ] Conditional pdf and cdf Suppose Event C given, P[C] > 0 (strictly positive) The conditional cd???? given C is: ???? ????|???? = ) ????[{????≤????}∩????] ???? ????[????] And ???? ????|???? )= ???? ????????(|???? a)plies ???????? Example: A lifeti????ehas cd???????????? distribution of lifetime Find the conditional cdf and pdf giC= {????v> t} Event that ????[{????≤????}∩{????>????}] machine lives at ????????????|???? > ???? ≡ ) ????[{????>????}] least this long The intersection of the two events in numerator = 0. 0 ???? ≤ ???? Therefore????????????|???? > ???? = { ????????(???? − ???? ???? ) 1−???????? ???? ) ???? > ???? In Summary: the derivative of the cdf is a pdf!! Lastly, the theorem on total probability allows us to find cdf of ???? ???? ???????????? = ???? ???? ≤ ???? = ] ∑ ???? ???? ≤ ???? ???? ????????????[ ] ???? ????=0 ???? = ∑????=0 ????????????|???? ???????????? [ ]????

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