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

• Notes
• 16

This preview shows page 4 - 7 out of 16 pages.

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 )

Subscribe to view the full document.

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 .
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 ) .

Subscribe to view the full document.

You've reached the end of this preview.
• Fall '08
• Stites,M
• Probability theory, lim

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern