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

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Unformatted text preview: Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University The Notion of a Random Variable A random variable X is a function that assigns a real number X ( ) to each outcome in the sample space of a random experiment. S is the domain, and S X = { X ( ) : S } is the range of r.v. X Example: toss a coin S = { H,T } X ( H ) = 0 ,X ( T ) = 1 ,S X = { , 1 } P ( H ) = P ( X = 0) = 0 . 5 ,P ( T ) = P ( X = 1) = 0 . 5 measure the temperature, S = { | 10 < < 30 } X ( ) = , then S X = { x | 10 < x < 30 } What is P ( X = 25)? ENCS6161 p.1/47 Cumulative Distribution Function F X ( x ) = P ( X x ) = P ( : X ( ) x )- < x < Properties of F X ( x ) 1. 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 > F X ( b ) = lim h F X ( b + h ) = F X ( b + ) (see examples below) ENCS6161 p.2/47 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 P { b- < X b } = F X ( b )- lim 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 ( ) = F X ( x ) = x < 1 1 6 1 x < 2 2 6 2 x < 3 . . . 5 6 5 x < 6 1 x 6 1 F X ( x ) F X ( x ) 1 2 3 4 5 6 x check properties 5,6,7. ENCS6161 p.4/47 Cumulative Distribution Function Example: pick a real number between and 1 uniformly F X ( x ) = x < x/ 1 0 x 1 1 x > 1 1 F X ( x ) 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 ,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 x < 1 x 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 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 < 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....
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This note was uploaded on 01/15/2011 for the course ECE 6161 taught by Professor Khkjk during the Winter '10 term at Concordia Canada.

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

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