Transcript PowerPoint Slides for Section 4.3 - Ursinus College Student, Faculty

4.3 Random Variables
Quantifying data
Given a sample space, we are often
interested in some numerical property of
the outcomes. For example, if our
collection is college students, we may be
interested in their height. Or their weight
or their IQ or any other property which we
could somehow assign a number.
This is motivation for the idea of a random
variable.
A random variable is a rule (like a function)
that enables us to assign a number to
each outcome of a sample space. The
actual number associated with the
outcome is called the value.
We distinguish between a discrete and
continuous random variable.
Some examples
1.
2.
3.
4.
5.
Suppose I have a collection of books, and I
pick one book at random. Then my sample
space is just that same collection of books. A
discrete variable could be
The number of pages of the book.
The number of words in the book.
The number of pages of the last chapter.
0 if the book does not contain a preface, 1 if
the book does contain a preface.
1 if the book is between 1 and 100 pages, 2 if
the book is between 101 and 200 pages, etc.
Let’s look at an example in detail. I have 208 (math)
books in my office.
Suppose I wish to organize them according to “area of
math” i.e. my discrete variable is “area of math.”
Here is the breakdown: 28 algebra, 55 history, 20
number theory, 48 Calculus, 57 topology.
Probability
Algebra
History
Number
Theory
Calculus
Topology
.13
.26
.10
.23
.28
This is an example of a probability distribution.
The Idea
Consider our old go-to experiment of tossing a
fair cone two times. The sample space
S={HH, HT, TH, TT}. What is the probability of
obtaining each event as an outcome?
Any event in the sample space has equal
probability, 1/4, of being selected. There are 4
possible outcomes and so 1/4+1/4+1/4+1/4=1.
This observation almost seems silly.
Suppose now that the coin is “weighted” with a
60% chance of landing on a head and a 40%
chance of landing on a tail. We still have the
same sample space, but now the probabilities
are as follows: P(HH)=.36, P(TH)=P(HT)=.24,
and P(TT)=.16. But again .36+.24+.24+.16=1.
We thus define the specifications of the
probabilities associated with the various distinct
values of a discrete random variable to be a
discrete probability distribution. The probability
associated with the value x is denoted P(x).
Examples
Consider the
probability distribution
Find P(x=C) and
P(x=E).
P(x=A or x=B).
In a general discrete
distribution, P(x=a or
x=b)=P(a)+P(b).
x
P(x)
A
1/12
B
2/5
C
1/3
D
1/20
E
2/15
Probability Histograms
Here we plot each x value
as a bar with height P(x).
For the data set
Construct a probability
histogram.
This gives a good graphic
way of displaying your
probability distribution.
x
P(x)
0
1/10
1
2/5
2
3/10
3
1/5
Practice Problem
The following shows the
distribution of the number
of misspellings in a 250
word essay.
Find P(x=2), P(x=1 or
x=6), P(x<4), and P(x≤4).
Construct a probability
histogram.
x
P(x)
0
.05
1
.15
2
.2
3
.25
4
.2
5
.07
6
.05
7
.03
Discrete vs. Continuous
Consider the example above where our random variable
is the number of pages of a book. It is certainly possible
for the value of a particular book to be 243 or 244. But is
it possible to be 243.11?
In contrast, consider a collection of the days of the year
in 2008 in Collegeville, and let our random variable
assign to each day the temperature on that particular
day in Collegeville. Not only is it possible to obtain the
values 53 and 54, but one could technically obtain the
value 53.7, 53.78, 53.781, etc. depending on how
accurate our instruments are, at least in theory. But
there is no such thing as a book with one-quarter of a
page.
This leads us to the following definition.
A random variable is said to be continuous if it
can potentially take on any value in between any
two points where the random variable is defined.
A random variable is discrete if the values it can
potentially assume a sequence of isolated
points.
Examples: The sum of a roll of 4 dice.
Age in whole years.
Age as exact as possible at a certain point.
Normal Distributions
Our old friend the normal distribution is an
example of continuous probability
distribution.
For μ=10 and σ=2, find P(x<6), P(x>11 or
x<4) and P(x≤6).