2 if y ax b then e y ae x b e x acts kind of like an

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2. If Y = aX + b then E [ Y ] = aE [ X ] + b . E [ X ] acts kind of like an integral of X ( ω ) over Ω , weighted by P . One way that the expectation is expressed is E [ X ] = Z Ω X ( ω ) P ( ) = Z Ω XdP. An integral in this form is said to be a Lebesgue-Stieltjes Integral. Since X induces a probability P X on ( R , B ) , as we have observed we can also think of the probability space ( R , B , P X ) . We can write E [ X ] = Z R xP X ( dx ) where now X is the “identity” r.v. on the real line. We thus have two equivalent definitions: Z Ω X ( ω ) P ( ) = Z R xP X ( dx ) Back to properties: 1. If Y = g X then E [ Y ] = Z Ω ( g X )( ω ) P ( ) = Z R g ( x ) P X ( dx ) = Z R yP Y ( dy ) Pairs of random variables Ultimately, we will be dealing with infinite sequences of random variables. As steps along the way, we will examine carefully pairs of random variables, then vectors of random variables. On R 2 , the smallest σ -field of interest is B 2 , which is the smallest σ -field containing all of the rectangles. This is the Borel σ -field of R 2 . Definition 1 A bivariate random variable ( X, Y ) is a measurable mapping from , F ) to ( R 2 , B 2 ) . 2 That is, { ω Ω : ( X, Y )( ω ) B } ∈ F∀ B ∈ B 2 Note that two r.v.s X, Y on , F ) form a bivariate r.v. Definition 2 The joint or bivariate distribution of ( X, Y ) is P XY ( B ) = P ( { ω Ω : ( X, Y )( ω ) B } ) for B ∈ B 2 . 2 Definition 3 The joint c.d.f. of ( X, Y ) is defined as F XY ( a, b ) = P ( X a, Y b ) = P (( X, Y ) R a,b )
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ECE 6010: Lecture 2 – More on Random Variables 5 where R ab is the semi-infinite rectangle R a,b = { ( x, y ) R 2 : x a, y b } . 2 Properties of the joint c.d.f.: 1. lim a,b →∞ F X,Y ( a, b ) = 1 . 2. lim a →-∞ F X,Y ( a, b ) = 0 = lim b →-∞ F X,Y ( a, b ) . 3. lim a →∞ F X,Y ( a, b ) = F Y ( b ) , the marginal c.d.f. of Y . lim b →∞ F X,Y ( a, b ) = F X ( a ) , the marginal c.d.f. of X . 4. F XY ( a, b ) is continuous “from the northeast.” 5. F XY ( x, y ) is montonically increasing (or, more precisely, nondecreasing) in both variables. Any function with these properties is a legitimate c.d.f., and completely characterizes the family of joint c.d.f.s. Joint discrete r.v.s Definition 4 If X, Y are discrete r.v.s taking values in sets { x 1 , . . . } and { y 1 , . . . , } , re- spectively, then ( X, Y ) forms a discrete bivariate r.v. and its joint p.m.f. is defined by p XY ( a, b ) = P ( X = a, Y = b ) 2 Properties of p XY : 1. p XY 0 , and p XY ( a, b ) = 0 if a 6∈ { x 1 , . . . } or b 6∈ { y 1 , . . . } 2. i =1 j =1 p XY ( x i , y j ) = 1 . 3. F XY ( a, b ) = { x i ,y i } : x i a,y j b } p XY ( x i , y j ) 4. Marginals: P X ( x i ) = X j p XY ( x i , y j ) P Y ( y j ) = X i p XY ( x i , y j ) Joint continuous r.v.s Definition 5 X and Y are jointly continuous r.v.s if there is a function f XY : R 2 R 2 such that F XY ( a, b ) = Z b Z a f XY ( x, y ) dxdy for all ( a, b ) R 2 . The function f XY is called the joint p.d.f. of X and Y (when it exists). 2 Properties of joint p.d.f.: 1. f XY 0 . 2. R -∞ R -∞ f XY ( x, y ) dx, dy = 1 .
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ECE 6010: Lecture 2 – More on Random Variables 6 3. We can get the p.d.f. from the c.d.f: f XY ( x, y ) = 2 ∂x∂y F XY ( x, y ) .
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  • Fall '08
  • Stites,M
  • Probability theory, lim

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