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# ch34 - Chapter 3 4 Random Variables ENCS6161 Probability...

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Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University

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The Notion of a Random Variable A random variable X is a function that assigns a real number X ( ω ) to each outcome ω in the sample space of a random experiment. S is the domain, and S X = { X ( ω ) : ω S } is the range of r.v. X Example: toss a coin S = { H, T } X ( H ) = 0 , X ( T ) = 1 , S X = { 0 , 1 } P ( H ) = P ( X = 0) = 0 . 5 , P ( T ) = P ( X = 1) = 0 . 5 measure the temperature, S = { ω | 10 < ω < 30 } X ( ω ) = ω, then S X = { x | 10 < x < 30 } What is P ( X = 25)? ENCS6161 – p.1/47
Cumulative Distribution Function F X ( x ) Δ = P ( X x ) = P ( ω : X ( ω ) x ) - ∞ < x < Properties of F X ( x ) 1. 0 F X ( x ) 1 2. lim x →∞ F X ( x ) = 1 3. lim x →-∞ F X ( x ) = 0 4. F X ( x ) is nondecreasing, i.e., if a < b then F X ( a ) F X ( b ) 5. F X ( x ) is continuous from right, i.e., for h > 0 F X ( b ) = lim h 0 F X ( b + h ) = F X ( b + ) (see examples below) ENCS6161 – p.2/47

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Cumulative Distribution Function Properties of F X ( x ) 6. P ( a < X b ) = F X ( b ) - F X ( a ) for a b Proof: { X a } ∪ { a < X b } = { X b } { X a } ∩ { a < X b } = so P { X a } + P { a < X b } = P { X b } P { a < X b } = F X ( b ) - F X ( a ) 7. P { X = b } = F X ( b ) - F X ( b - ) Proof: P { X = b } = lim ε 0 P { b - ε < X b } = F X ( b ) - lim ε 0 F X ( b - epsilon1 ) = F X ( b ) - F X ( b - ) ENCS6161 – p.3/47
Cumulative Distribution Function Example: roll a dice S = { 1 , 2 , 3 , 4 , 5 , 6 } , X ( ω ) = ω F X ( x ) = 0 x < 1 1 6 1 x < 2 2 6 2 x < 3 . . . 5 6 5 x < 6 1 x 6 0 1 F X ( x ) 1 2 3 4 5 6 x check properties 5,6,7. ENCS6161 – p.4/47

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Cumulative Distribution Function Example: pick a real number between 0 and 1 uniformly F X ( x ) = 0 x < 0 x/ 1 0 x 1 1 x > 1 0 1 F X ( x ) 1 2 x P { X = 0 . 5 } =? ENCS6161 – p.5/47
Three Types of Random Variables 1. Discrete random variables S X = { x 0 , x 1 , · · · } finite or countable Probability mass function ( pmf ) P X ( x k ) = P { X = x k } F X ( x ) = k P X ( x k ) u ( x - x k ) where u ( x ) = braceleftBigg 0 x < 0 1 x 0 see the example of rolling a dice F X ( x ) = 1 6 u ( x - 1) + 1 6 u ( x - 2) + · · · + 1 6 u ( x - 6) ENCS6161 – p.6/47

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Three Types of Random Variables 2. Continuous random variables F x ( x ) is continuous everywhere P { X = x } = 0 for all x 3. Random variables of mixed type F X ( x ) = pF 1 ( x ) + (1 - p ) F 2 ( x ) , where 0 < p < 1 F 1 ( x ) CDF of a discrete R.V F 2 ( x ) CDF of a continuous R.V Example: toss a coin if H generate a discrete r.v. if T generate a continuous r.v. ENCS6161 – p.7/47
Probability density function PDF, if it exists is defined as: f X ( x ) Δ = d F X ( x ) d x f X ( x ) F X ( x x ) - F X ( x ) Δ x = P { x<X x x } Δ x "density" Properties of pdf (assume continuous r.v.) 1. f X ( x ) 0 2. P { a X b } = integraltext b a f X ( x ) d x 3. F X ( x ) = integraltext x -∞ f X ( t ) d t 4. integraltext + -∞ f X ( t ) d t = 1 ENCS6161 – p.8/47

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Probability density function Example: uniform R.V f X ( x ) = braceleftBigg 1 b - a a x b 0 elsewhere 1 b - a f X ( x ) a b x F X ( x ) = 0 x < a x - a b - a a x b 1 x > b 1 F X ( x ) a b x ENCS6161 – p.9/47
Probability density function Example: if f X ( x ) = ce - α | x | , -∞ < x < + 1. Find constant c integraldisplay + -∞ f X ( x ) d x = 2 integraldisplay + 0 ce - αx d x = 2 c α = 1 c = α 2 2. Find P {| X | < v } P {| X | < v } = α 2 integraldisplay v - v e - α | x | d x = 2 α 2 integraldisplay v 0

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ch34 - Chapter 3 4 Random Variables ENCS6161 Probability...

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