ECE313.Lecture32

# ECE313.Lecture32 - ECE 313 Probability with Engineering...

This preview shows pages 1–8. Sign up to view the full content.

Jointly Continuous Random Variables I Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 36 Probability mass on the real line The pmf p X (u) for a discrete random variable X describes a collection of point masses on the real line: P{ X = u} = p X (u) A continuous random variable X spreads the mass on (an interval of) the real line There is no probability mass at any point P{ X = u} = 0 for all real numbers u pdf f X (u) is the density of the probability mass Units are probability mass per unit length P{ X { small interval that contains number u}} ≈ f X (u)•{length of the interval}
ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 36 Joint behavior of continuous RVs When X and Y are discrete RVs, the random point ( X , Y ) is also discrete valued The joint pmf p X , Y (u,v) describes point masses in the plane If X and Y are continuous RVs, then either ( X , Y ) can take on all possible values in a region of nonzero area or Y = g( X ), and thus ( X , Y ) always lies on the curve v = g(u) in the plane

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 36 Probability mass in the plane Suppose that X and Y are continuous RVs, and the random point ( X , Y ) can take on all values in a region of nonzero area In this case, X and Y are said to be jointly continuous RVs The probability mass is spread over this region of the plane If Y = g( X ), then the probability mass is spread along the curve v = g(u) X and Y are not jointly continuous RVs
ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 36 Probability mass along a curve If Y = g( X ), then the probability mass is spread along the curve v = g(u) X and Y are not jointly continuous RVs All questions involving the probabilistic behavior of the random point ( X , Y ) can be translated into questions involving X alone Example: If Y = X 2 , then for any v ≥ 0, the joint CDF of X and Y is: F X , Y (u,v) = P{ X ≤ u, Y ≤ v} = P{ X ≤ u, X 2 ≤ v} = P{–a ≤ X ≤ a} where a = min{u, v}

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
ECE 313 - Lecture 32 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 36 Probability mass over a region The case of X and Y being continuous random variables but not jointly continuous random variables is easily treated via the methods that we have studied previously The interesting case is when X and Y are jointly continuous and the probability mass is spread over a region of the plane Abuse of language: We will assume that, unless explicitly stated otherwise, continuous RVs are also jointly continuous

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.

### Page1 / 38

ECE313.Lecture32 - ECE 313 Probability with Engineering...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online