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# Week 6 Notes ELEG310

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This 10 page Class Notes was uploaded by Cindy Nguyen on Saturday March 19, 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 25 views. For similar materials see Random Signals and Noise in Electrical Engineering at University of Delaware.

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Date Created: 03/19/16

ELEG 310 Lectures 9 and 10 3/19/16 Topic 1: Expected Value Consider a distribution with a de????????????ty of Suppose ∆ is a small number, small interval size. If we do n experiments, we expect ???? ???? to come out in inte???? ???? ∆) ???? ???? Expectation Inspiration behind definition: E[???????? ???? ????(????????)∆ ∞ Actual definition: E[????∫−∞ ???? ???? (????)???????? E[????] = ∞ ????[∑ ???? ???? ???? ???? − ???? ]???????? ∫−∞ ???? ???? ???? 0 Switch sum and integral E[????] = ∑ ???? ???? )∫ ∞ ???? ???? ???? − ???? ???????? ???? ???? ???? −∞ 0 E[????] = ∑???? ???? ???? ????)???????? Example 1: a continuous interval from a to b: [a,b] Plug in; integrate pdf*variable. The pdf is a constant E[????] = 1 ∫???? ???? ???????? ????−???? ???? 2 = 1 [ ] ???? ????−???? 2 ???? 2 2 = ???? −???? 2(????−????) = ????+???? 2 Example 2: pdf of an exponential random variable ???? ???? = ???????? −???????? ???? ∞ E[????] = ∫ ???? ???????????????? ???????? 0 Re-write as = 1∫ ∞ ???? ???????????????? ???? ???????? ( ∗ ???? ???????????????? ???????????????????????? 1) ???? 0 ???? This makes it easier to use integration by parts. Sub x = ???????? 1 ∞ −???? −???? = ????∫0 ???????? ???????? where u = x and dv = ???? 1 −???? ∞ ∞ −???? = ????−???????? ] 0+ ∫0 ???? ???????? 1 −???? ∞ = ????−???? ]0 1 E[????] = ???? Example 3 Let ???? = ????(????) ∞ Then ???? ???? = ∫−∞ ????(????) ???? ????( ) It is distributed uniformly in the interval [a,b] 1 ???? E ???? 2]= ∫???? ???? ???????? ????−???? 3 = 1 [ ] ???? ????−???? 3 ???? 3 3 = ???? −???? 3(????−????) ???? −????3 = 3(????−????) (????−????)(???? +2????????+???? ) = 3(????−????) (???? +2????????+???? ) = 3 1 2 2 = 3(???? + 2???????? + ???? ) Example 4 Sine Function Sin ???? ???? ∈ [0,????] E [Sin(????)] = 1∫???? sin ???? ???????? ???? 0 1 = [−cos ???? ] ) ???? ???? 0 1 = [1 − (−1)] ???? 2 = ???? Example 5 Variance over interval [a,b] Recall: Variance = VAR[????] = ???? ????] − (???? ???? ) 2 1 2 2 ????+???? 2 = 3(???? + 2???????? + ???? ) − ( 2 ) E ???? 2]???????????????? ???????????????????????????? 3 2 (???? ???? ) ???????????????? ???????????????????????????? 1 (????−????)2 = 12 Physics connection: equation for inertia of a rod Example 6 Variance of an Exponential Random Variable Let E ????2] = ∫ ∞ ???? ????????−???????????????? 0 Must do integration by parts, twice. 1 ∞ 2 −???????? ????2 = ????2∫0 (???????? ???? (????????????) multiply by????2for ease 1 ∞ 2 −???? = ????2∫0 ???? ???? (????????) variable change x=???????? for ease Parts: u = x2 v = -ex -x du =2xdx dv = e dx 1 ∞ = 2[ −???? ????−???? ]∞ + 2 ∫ ????????−????(????????) ???? 0 0 2 E ???? 2]= 2 ???? VAR[????] = ???? ????[ 2]− (???? ???? ) 2 [ ] 2 1 2 (E ???? ) = (????) (from example 2) Topic 2: Properties of Variance Variance of a constant Recall: E[C] = C E[???? +C] = E[????] + C VAR[C] = E[C ] – E[C] 2 2 = C – C = 0 This makes sense because a constant doesn’t vary. No variance Variance of constant times random variable VAR[C ????] = E[C ???? ] − E[C????] = C E[???? ] − C(E[????])2 Pull out C 2 2 2 = C (E[???? ] − E[????] ) = C (VAR[????]) Topic 3: Continuous Random Variables 1 ????−???? ????∈[????,????] Density ???? = { ????????ℎ???????????????????????? 0 0 ???? < ???? ????−???? ???????????? = {????−???? ???? ∈ ????,???? ] (in between a and b it must rise linearly) 1 ???? > ???? ????+???? E[u] = 2 (????−????)2 VAR[u] = 12 (from Example 5, page 3) 0 ???? < 0 ???????????? = { −???????? ???????? ???? > 0 0 ???? < 0 ???? ???? = { 1 − ???? −???????? ???? > 0 Topic 4: Suppose the lifetime of a machine is x ???? ???? > ???? + ℎ ???? > ????] = ????[???? > ℎ] It doesn’t matter Time “h” longer what t is because it’s memoryless By definition, ????[ ????>????+ℎ ∩ ????>???? ] ???? ???? > ???? + ℎ ???? > ????] = ????[ ????>???? ] ????[ ????>????+ℎ ] = ????[ ????>???? ] Topic 5: Gaussian (a.k.a. “Normal”) Random Variable (????−????) 1 − 2 ???????????? = ???? 2???? ???? 2???? where m = mean, ????=standard deviation √ This is a “bell curve”. 2 1 ∞ −(????−????) ???????????? = ∫ [???? 2????2 ]???????? this integral can’t be done ????√2???? −∞ ′ So, substitute t =???? −???? and dt = ????????′ ???? ???? ′ 1 ???? −???? − ???? ????−???? ???????????? = ∫ ???? [???? ????]???????? = Φ ????√2???? −∞ ???? ELEG310 Lecture 10 Linear Function ????[???? ∈ ????] Y in some set C ???? = ????(????) ???? ???? ∈ ???? = ???? ???? ???? ∈ ???? )] = ????[???? ∈ ???? (????)] ????−1 (???? = {????:????(????) ∈ ????} ???? = ???????? + ???? ???? > 0 ???? ???????????? ???? ???????????? ???????????????????????? ???????????????????????????????????? ???? ???????????? ???? ???????????? ???????????????????????????????????? ???? ???? = ???? ???? ≤ ???? ] ???? = ???? ???????? + ???? ≤ ???? ] = ???? ???????? ≤ ???? − ???? ] ????−???? = ????[???? ≤ ???? ] ????−???? = ????????( ???? ) Suppose a < 0 ????[???????? ≤ ???? − ????] = ???? ????( ) ????−???? ????−???? ????[???? ≥ ???? ] = 1 − ????????( ???? ) ????−???? For ????????= ( ) ,???? > 0 ???? ????−???? 1 ???????????? = ???? (???? )( ) ???? ???? 1 ????−???? = ( )????????( ) ???? ???? For ???? < 0, 1 ????−???? ???? ???? = − ???? ???? ???? ???? ) However, whether a is positive or negative, this equation works, for linear functions. 1 ????−???? ???? ???? = ????????( ) |????| ???? Chebyshev Inequality ????2 ???? ???? − ???? ≥ ???? ≤ ] ????2 2 2 Proof: Let ???? = (???? − ????) 2 ???? ???? ≥ ???? 2] ≤ ????[???? ] ????2 ????[????−???? ] = ????2 ???? 2 = ????2 Bermoulli random variable, with a fair coin 1 ???? ???? = 0 = 2 1 ???? ???? = 1 = 2 [ ] 1 ???? ???? = 2= ???? ???????????? ???? = 1 4 ???? ???? 2]= 0 2 ( )+ 1 2 ( ) 2 2 1 = 2 1 1 ???????????? ???? = − ( )2 2 2 = 1 4 1 1 1 4 ????[|???? − |2≥ ] 2 1= 1 4

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