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

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

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

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?”

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)

ECE 313 - Lecture 7 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved