[PPT] - EE361: SIGNALS AND SYSTEMS II CH2: RANDOM VARIABLES PowerPoint Presentation

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http://www.ee.unlv.edu/~b1morris/ee361

EE361: SIGNALS AND SYSTEMS II

CH2: RANDOM VARIABLES

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INTRODUCTION

 A Random Variable is a function that maps an

event to a probability (real value)

 Will use distribution functions to describe the

functional mapping

 Example: your score on the midterm is a random

variable and the Gaussian distribution explains the probability you achieved a certain value (e.g. 70/100)

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 𝑌(𝜊) is a single-valued real

function that assigns a real number (value) to each sample point (outcome) in a sample space 𝑇

 Often just use 𝑌 for simplicity  This is a function (mapping) from

sample space 𝑇 (domain of 𝑌) to values (range)

 This is a many-to-one mapping

 Different 𝜊𝑗 may have same value 𝑌(𝜊𝑗),

but two values cannot come from same

utcome

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RANDOM VARIABLE

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EVENTS DEFINED BY RVS

 Event

 𝑌 = 𝑦 = 𝜊: 𝑌 𝜊 = 𝑦  RV 𝑌 value is 𝑦, a fixed real number

 Similarly,

 𝑦1 < 𝑌 ≤ 𝑌2 = 𝜊: 𝑦1 < 𝑌 𝜊 ≤ 𝑦2

 Probability of event

 𝑄 𝑌 = 𝑦 = 𝑄 𝜊: 𝑌 𝜊 = 𝑦

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EXAMPLE: COIN TOSS 3 TIMES

 Sample space 𝑇 = 𝐼𝐼𝐼, 𝐼𝐼𝑈, … , 𝑈𝑈𝑈 , 𝑇 = 23 = 8  Define RV 𝑌 as the number of heads after the three tosses  Find 𝑄(𝑌 = 2)

 Event A: 𝑌 = 2 = 𝜊: 𝑌 𝜊 = 2 = {HHT, HTH, HTT}  By equally likely events

 𝑄 𝐵 = 𝑄 𝑌 = 2 =

𝐵 𝑇 = 3 8

 Find 𝑄(𝑌 < 2)

 Event B: 𝑌 < 2 = 𝜊: 𝑌 𝜊 < 2 = HTT, THT, HTT, TTT (1 or less heads)  By equally likely events

 𝑄 𝐶 = 𝑄 𝑌 < 2 =

𝐶 𝑇 = 4 8 = 1 2

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CUMULATIVE DISTRIBUTION FUNCTION (CDF)

 𝐺

𝑌 𝑦 = 𝑄 𝑌 ≤ 𝑦

−∞ < 𝑦 < ∞

 𝐺 – the CDF  𝑌 – the RV of interest  𝑦 – the value the RV will take

 Note: this is an increasing (non-decreasing) function

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CDF PROPERTIES

 1) 0 ≤ 𝐺

𝑌 𝑦 ≤ 1

 Must be less than some maximal value

 2) 𝐺

𝑌 𝑦1 ≤ 𝐺 𝑌(𝑦2)

if 𝑦1 < 𝑦2

 Non-decreasing function

 …  5) lim𝑦→𝑏+ 𝐺

𝑌 𝑦 = 𝐺 𝑌 𝑏+ = 𝐺 𝑌 𝑦

with 𝑏+ = lim0<𝜗→0 𝑏 + 𝜗

 Continuous from the right

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EXAMPLE: 3 COIN TOSS AGAIN

 𝑌 – number of heads in three tosses

8 𝒚 (value) Event (𝒀 ≤ 𝒚) # elements 𝑮𝒀(𝒚)

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∅ {TTT} 1 (1 + 0) 1 8 1 2 3 4

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EXAMPLE: 3 COIN TOSS AGAIN

 𝑌 – number of heads in three tosses

9 𝒚 (value) Event (𝒀 ≤ 𝒚) # elements 𝑮𝒀(𝒚)

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∅ {TTT} 1 (1 + 0) 1 8 1 {HTT, THT, TTH, TTT} 4 (3+1) 4 8 = 1 2 2 3 4

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EXAMPLE: 3 COIN TOSS AGAIN

 𝑌 – number of heads in three tosses

10 𝒚 (value) Event (𝒀 ≤ 𝒚) # elements 𝑮𝒀(𝒚)

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∅ {TTT} 1 (1 + 0) 1 8 1 {HTT, THT, TTH, TTT} 4 (3+1) 4 8 = 1 2 2 {HHT, HTH, THH, HTT, THT, TTH, TTT} 7 (3 + 4) 7 8 3 4

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EXAMPLE: 3 COIN TOSS AGAIN

 𝑌 – number of heads in three tosses

11 𝒚 (value) Event (𝒀 ≤ 𝒚) # elements 𝑮𝒀(𝒚)

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∅ {TTT} 1 (1 + 0) 1 8 1 {HTT, THT, TTH, TTT} 4 (3+1) 4 8 = 1 2 2 {HHT, HTH, THH, HTT, THT, TTH, TTT} 7 (3 + 4) 7 8 3 {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 8 (1 + 7) 1 4 𝑇 8 (0 + 8) 1

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EXAMPLE: 3 COIN TOSS AGAIN

 𝑌 – number of heads in three tosses

12 𝒚 (value) Event (𝒀 ≤ 𝒚) # elements 𝑮𝒀(𝒚)

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∅ {TTT} 1 1 8 1 {HTT, THT, TTH, TTT} 4 (3+1) 4 8 = 1 2 2 {HHT, HTH, THH, HTT, THT, TTH, TTT} 7 (3 + 4) 7 8 3 {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 8 (1 + 7) 1 4 𝑇 8 (0 + 8) 1

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PROBABILITIES FROM CDF

 Completely specify probabilities from a CDF  1) 𝑄 𝑏 < 𝑌 ≤ 𝑐 = 𝐺

𝑌 𝑐 − 𝐺 𝑌 𝑏

= 𝑄 𝑌 ≤ 𝑐 − 𝑄(𝑌 ≤ 𝑏)

 2) 𝑄 𝑌 > 𝑏 = 1 − 𝐺

𝑌 𝑏

 3) 𝑄 𝑌 < 𝑐 = 𝐺

𝑌 𝑐−

 b− = lim0<𝜗→0 𝑐 − 𝜗  Approach from the left side

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DISCRETE RV

 𝑌 is RV with CDF 𝐺

𝑌 𝑦 and 𝐺 𝑌(𝑦) only changes in

jumps (countably many) and is constant between jumps

 Range of 𝑌 contains a finite (countably infinite) number

f points

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PROBABILITY MASS FUNCTION (PMF)

 Given jumps in discrete RV @ points 𝑦1, 𝑦2, … and 𝑦𝑗 < 𝑦𝑘

for 𝑗 < 𝑘

 𝑞𝑌 𝑦 = 𝐺

𝑌 𝑦𝑗 − 𝐺 𝑌 𝑦𝑗−1

= 𝑄 𝑌 ≤ 𝑦𝑗 − 𝑄 𝑌 ≤ 𝑦𝑗−1 = 𝑄 𝑌 = 𝑦𝑗

 3 Coin toss example

15 𝒚 (value) # elements 𝑮𝒀(𝒚) 𝒒𝒀(𝒚) Discussion 1 4 (3+1) 4 8 = 1 2 𝑞𝑌 1 = 4 8 − 1 8 = 3 8 <how much more needed from previous value> 2 7 (3 + 4) 7 8 𝑞𝑌(2) = 7 8 − 1 2 = 3 8 3 extra outcomes 3 8 (1 + 7) 1 𝑞𝑌 3 = 1 − 7 8 = 1 8 1 extra outcome

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PMF PROPERTIES

 1) 0 ≤ 𝑞𝑌 𝑦𝑙 ≤ 1

𝑙 = 1, 2, … (finite set of values)

 2) 𝑞𝑌 𝑦 = 0 if 𝑦 ≠ 𝑦𝑙 (a value that cannot occur)  3) σ𝑙 𝑞𝑌(𝑦𝑙) = 1  CDF from PMF

 𝐺

𝑌 𝑦 = 𝑄 𝑌 ≤ 𝑦 = σ𝑦𝑙≤𝑦 𝑞𝑌(𝑦𝑙)

 Accumulation of probability mass

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CONTINUOUS RV

 𝑌 is RV with CDF 𝐺

𝑌 𝑦 continuous and has a

derivative

𝑒𝐺𝑌 𝑦 𝑒𝑦

exists

 Range contains an interval of real numbers

 Note: 𝑄 𝑌 = 𝑦 = 0

 There is zero probability for a particular continuous

utcome  only over a range of values

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 𝑔

𝑌 𝑦 = 𝑒𝐺𝑌 𝑦 𝑒𝑦

pdf of 𝑌

 Properties  1) 𝑔

𝑌 𝑦 ≥ 0

 2) ׬

−∞ ∞ 𝑔 𝑌 𝑦 𝑒𝑦 = 1

 3) 𝑔

𝑌 𝑦 is piecewise continuous

 4) 𝑄 𝑏 < 𝑌 ≤ 𝑐 = ׬

𝑏 𝑐 𝑔 𝑌 𝑦 𝑒𝑦

= 𝑄 𝑏 ≤ 𝑌 ≤ 𝑐 = 𝐺

𝑌 𝑐 − 𝐺 𝑌(𝑏)

 CDF from PDF

 𝐺

𝑌 𝑦 = 𝑄 𝑌 ≤ 𝑦 = ׬ −∞ 𝑦 𝑔 𝑌 𝜊 𝑒𝜊 18

PROBABILITY DENSITY FUNCTION (PDF)

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MEAN

 Expected value of RV 𝑌  Discrete

 𝜈𝑌 = 𝐹 𝑌 = σ𝑙 𝑦𝑙𝑞𝑌(𝑦𝑙)

 Continuous

 𝜈𝑌 = 𝐹 𝑌 = ׬

−∞ ∞ 𝑦𝑔 𝑌 𝑦 𝑒𝑦

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MOMENT

 nth moment defined as  Discrete

 𝐹 𝑌𝑜 = σ𝑙 𝑦𝑙

𝑜 𝑄 𝑌 𝑦𝑙

 Continuous

 𝐹 𝑌𝑜 = ׬

−∞ ∞ 𝑦𝑜𝑔 𝑌 𝑦 𝑒𝑦

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 𝜏𝑌

2 = 𝑊𝑏𝑠 𝑌 = 𝐹

𝑌 − 𝐹 𝑌

2

 𝐹[. ] – expected value operation  𝐹 𝑌 = 𝜈𝑌 - mean

 Discrete

 𝜏𝑌

2 = σ𝑙 𝑦 − 𝜈𝑌 2𝑞𝑌(𝑦𝑙)

 Continuous

 𝜏𝑌

2 = ׬ −∞ ∞

𝑦 − 𝜈𝑌 2𝑔

𝑌 𝑦 𝑒𝑦 21

VARIANCE