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ECE313.Lecture07

# ECE313.Lecture07 - ECE 313 Probability with Engineering...

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Mean, LOTUS, and Variance 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 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 40 Review of random variables A random variable X associates a number with each outcome of an experiment A random variable is a fixed map from the sample space to the real line Random because we do not know exactly which outcome of the experiment will be observed on the next trial, and thus which value the random variable will have Observed value of X varies at random
ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 40 Discrete random variables A discrete random variable takes on a finite number or a countably infinite number of discrete values The values taken on by a discrete random variable are discretely spaced If u 1 , u 2 , … are the values taken on by a discrete random variable, then for each choice of j, u j < u j+1

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ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 40 Probability mass functions (pmfs) The probabilistic behavior of a discrete random variable X is described by its probability mass function p X (u) or p(u) p X (u) = p(u) = 0 unless u is one of the values u j that X takes on p X (u j ) = p(u j ) = P{X = u j } p(u) ≥ 0 for all u; p(u j ) = 1 j
ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 40 Everything you always wanted to know about X All the probabilistic information about the discrete random variable X is summarized in its pmf The pmf can be used to answer questions such as “What is the probability that X has value between a and b ?” “What is the probability that X is an even number?”

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ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 40 … but were afraid to ask! Example: X is a random variable taking on integer values 0 through 8 p X (0) = p X (1) = p X (7) = p X (8) = 0.05 p X (2) = p X (3) = p X (5) = p X (6) = 0.15; p X (4) = 0.2 P{3 < X < 6} = p X (4) + p X (5) = 0.35 P{3 ≤ X < 6} = p X (3) + p X (4) + p X (5) = 0.5 P{ X is odd} = p X (1) + p X (3) + p X (5) + p X (7) = 0.4 P{ X = 3.13} = 0 because p X (3.13) = 0
ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 40 I’m leaving on a jet plane … Example (continued): Suppose that X is the number of passengers (with confirmed reservations) who show up for a ﬂight on a 5-passenger plane. Let Y denote the number of passengers who board the ﬂight Then, Y = X if X ≤ 5 and Y = 5 if X > 5 The random variable Y is said to be a function of the random variable X The pmf of Y can be found from p X (u)

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