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Chapter 16
Random Variables
Random Variables
These are variables which are extracted
from random phenomena.
 Let’s say we wanted to know the number
of heads found on 3 flips of a fair coin.
 HTH and THH are two different
outcomes.
 They are, however, the same number of
heads.

Random Variables

So, to clarify, the outcomes are converted into
the number of heads, and the number of
heads is what we consider on this random
variable.
 We get the following chart:
# of Heads
Probability
0
.125
1
.375
2
.375
3
.125
What is the point?
Random variables refer to numbers
taken from random events.
 These can be the outcomes, or else
counts of outcomes.
 Now that these are numbers, though,
instead of categorical, we can do even
more calculations with them.

Mean and Standard Deviation
Woohoo!
 Seriously, though, mean and standard
deviation of a random variable are easy.
 The formulas look intimidating, but it is a
smokescreen.

Mean





Now we will also called it expected value.
The expected value is the mean.
The mean is the expected value.
The mean may still be written µ, but now it
might also be written E(X) or E(Whatever), for
the expected value.
To find expected value, we multiply each
outcome by its probability and add them all up.
Example Time!

If we have the
following chart:
 0 x .5 = 0, 1 x .3 = .3,
2 x .15 = .3, 3 x .05 =
.15.
 0 + .3 + .3 + .15 =
.75
 So the mean is .75
X
0
1
2
3
P(X)
.5
.3
.15
.05
Variance





The standard deviation is the square root of
the variance.
We will find the variance first.
Then, we will square root that answer if we
want the standard deviation.
The formula is almost as easy.
We will multiply the square of each outcome
with the probability, add them up, and then
subtract the mean squared.
Example Time

We will use our
previous example.
 The mean was .75.
 In order to find the
variance, we will
need to square each
outcome, or each X
value.
X
0
1
2
3
P(X)
.5
.3
.15
.05
Example Time






So our new table looks
like this:
So 0 x .5 = 0, 1 x .3 =
.3, 4 x .15 = .6, 9 x .05
= .45.
0 + .3 + .6 + .45 = 1.35
.752 = 0.5625
1.35 – 0.5625 =
0.7875
The variance is 0.7875
X
0
1
2
3
P(X)
.5
.3
.15
.05
X2
0
1
4
9
Example Time

The standard deviation is the square root of
the variance, so:
  .7875  08874
.
Mean and Standard Deviation
It is seriously that easy.
 Not even kidding.
 The book uses a harder formula for
variance, so feel free to use that one
instead.
 Hehehe.

Mean and Variance
If we take a random variable and multiply
it by a number, we can find the new
mean by multiplying the old mean by that
number.
 We multiply the standard deviation by
that number.
 We multiply the variance by the square
of that number.

Mean and Variance
When we add two random variables, we
add their means and we add their
variances.
 When we subtract two random variables,
we subtract their means and we add
their variances.
 That’s right! We add variances in both
cases.

Mean and Variance

If we know the standard deviation instead of
the variance, we must square it to get the
variance.
 Only after we have squared can we add the
variances together.
 Let’s look at some examples!
 For those of you that are not here, you will
miss these examples, as they are being done
on the board.
Super Example Time Activate!

First, we will use this
data to find the mean
and standard
deviation.
 Then we will focus
on the variance as
well.
X
0
1
2
3
P(X)
.5
.3
.15
.05
Super Example Time Activate!

Find the mean and standard deviation for
the following things:
3X
 X+X
 X-X
 5X+3X
 5X-3X

Assignments

Chapter 15 homework due tomorrow.

Problems 10, 20, and my special problem.
Finish reading chapter 16.
 Chapter 16: 15, 25, 32
