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# More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

## 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 , the smallest -field of interest is , which is the smallest -field containing all of the rectangles. This is the Borel -field of .

That is,

Note that two r.v.s on form a bivariate r.v.

Properties of the joint c.d.f.:

1. .
2. .
3. , the marginal c.d.f. of .

, the marginal c.d.f. of .

4. is continuous ''from the northeast.''
5. 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.

Properties of :

1. , and if or
2. .
3. Marginals:

## Joint continuous r.v.s

Properties of joint p.d.f.:

1. .
2. .
3. We can get the p.d.f. from the c.d.f:

4. Marginals:

If and are jointly continuous, then they are marginally continuous (that is, and are continuous). However, the opposite is not true.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec2_3.html. This work is licensed under a Creative Commons License