{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ECE313.Lecture24

# ECE313.Lecture24 - ECE 313 Probability with Engineering...

This preview shows pages 1–9. Sign up to view the full content.

Continuous Random Variables III 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 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 The pdf of a continuous RV The pdf f X (u) of a continuous random variable is the derivative of its CDF The pdf has two properties f X (u) ≥ 0 Total area under the pdf curve f X (u) from to is 1 Fancy statement: f X (u) du = 1
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Other properties enjoyed by the pdf The pdf is a nonnegative function that has unit area between the pdf curve and the horizontal axis f X (+ ) = lim u + f X (u) = 0 f X (– ) = lim u f X (u) = 0 Compare to F X (+ ) = 1 and F X (– ) = 0

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

View Full Document
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Probabilities from the pdf P{a ≤ X ≤ a + δ a} ≈ f X (a)• δ a P{a < X < b} = Area under the pdf curve between a and b a b = f X (u) du P{ X = u} = 0 for all real numbers u P{a < X < b}, P{a < X ≤ b}, P{a ≤ X < b}, and P{a ≤ X ≤ b} all have the same value
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 Relationship between CDF and pdf The value of the CDF at the point u = 5 is F X (5) = area under pdf f X (u) from – to 5 The CDF is not the antiderivative (or indefinite integral) of the pdf; it is the definite integral Re-read your calculus books to refresh your understanding of the Fundamental Theorem of Calculus

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

View Full Document
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 ECE 313 Survival Guide Always, always, always sketch the pdf (or CDF) curve before you do anything else Indicate the desired probability as an area (e.g. by shading) on the sketch If you use an integral to find the area, set the limits with the help of the sketch Do not use indefinite integrals All integrals must have limits
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 Expectation of an arbitrary RV u 1 E[ X ] = [1 – F X (u)] du F X (u) du 0 0 The expected value E[ X ] of an arbitrary random variable X can be defined as F X (u) = blue area – orange area

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

View Full Document
ECE 313 - Lecture 24 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 8 of 40 Expectation of a continuous RV E[ X ] = [1 – F X (u)] du F X (u) du 0 0 E[ X ] = u•f X (u) du
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 41

ECE313.Lecture24 - ECE 313 Probability with Engineering...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online