ECE313.Lecture11

ECE313.Lecture11 - ECE 313 Probability with Engineering...

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Important Counting Random Variables 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
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Review: binomial random variables X denotes the number of occurrences of an event A of probability p on n trials X is called a binomial random variable with parameters (n, p) X takes on values 0, 1, 2, … n For 0 ≤ k ≤ n, Mean E[ X ] = np, variance np(1–p), and mode (n+1)p p X (k) = P{ X = k} = p k (1 – p) n–k n k
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Just so’s there’s no confusion … Binomial random variables are one class of the important discrete random variables that are the subject of this lecture Memorize the basic information shown on the previous slide … or at least have it on your sheet(s) of notes on the exams! On exams, you are expected to have this basic information at your fingertips
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 What if n is large and p is small? Suppose that n is large and p is very small The pmf of the binomial random variable X can be approximated and expressed as a function of n•p, that is, we do not need to know the values of n and p separately ; their product is all that is required λ = n•p It is assumed that λ is of moderate size What’s this large, small, moderate stuff?
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 n is large, p is small , λ is moderate? And Baby Bear’s chair was just right, too? The approximations that we will develop work reasonably well even for relatively small values of n, e.g. n ≈ 100 n large, p small and λ = n•p is moderate means p << 1 and λ << n For n ≈ 100, a λ of 15 or less will give reasonably accurate approximations
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 When all else fails, approximate … Remember np = λ . Then, for k << n, Consider the ratio containing k terms p X (k) = P{ X = k} = p k (1 – p) n–k n k n(n–1)(n–2)… (n–k+1) λ n 1 × 2 × 3 × × k n = 1 – p
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ECE 313 - Lecture 11 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 Aargh! Calculus rears its ugly head! Remember np =
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ECE313.Lecture11 - ECE 313 Probability with Engineering...

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