Download 7.2 Day 1: Mean & Variance of Random Variables

7.2 Day 1: Mean & Variance of
Random Variables
Law of Large Numbers
The Mean of a Random Variable

The mean x of a set of observations is their
ordinary average, but how do you find the
mean of a discrete random variable whose
outcomes are not equally likely?
The Mean of a
Random
Variable is
known as its
expected value.
Ex 1: The Tri-State Pick 3

In the Tri-State Pick 3 game that New
Hampshire shares with Maine and Vermont,
you choose a 3-digit number and the state
chooses a 3-digit winning number at random
and pays you $500 if your number is chosen.
Since there are 1000
possible 3 digit
numbers, your
probability of winning
is 1/1000.
The probability distribution of X
In the long run, you
(the amount your ticket payswould
you)
only receive
Payoff X:
Probability:

$0
0.999
$500 once in every
1,000 tickets and $0 in
the remaining 999 of
the tickets
$500
0.001
The ordinary average of the two possible
outcomes is $250, but that makes no sense
as the average because $0 is far more likely
than $500.
So what is the mean?
We will say
that
μx = $0.50.
1
999
$500
 $0
 $0.50
1000
1000


The long-run average payoff or mean for this
random variable X is fifty cents.
This is also known as the Expected Value.
Mean of a Discrete Random Variable

Suppose that X is a discrete random variable
whose distribution is
Value of X: x1 x2 x3 … xk
Probability: p1 p2 pWe
… usepμk to
3 will
x
signify that this is
 To find the mean of X, multiply each
possible
the mean
of a
random
variable
vlaue by its probability, then add
all the
products
and not of a data
 μx = x1p1 + x2p2 + … + xkpkset.

= Σxipi
Ex 2: Benford’s Law
Calculating the expected first digit

What is the expected
of the first digit if
The value
expected
each digit is equally
likely?
value
is μ = 5.
x
First Digit X
1
2
3
4
5
6
7
8
9
Probability
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
1/9
μx = 1(1/9) + 2(1/9) + 3(1/9) + 4(1/9) + 5(1/9) + 6(1/9) +
7(1/9) + 8(1/9) + 9(1/9)
=5

What is the expected value if the data obeys
Benford’s Law?
First Digit X
Probability
1
.301
2
3
4
5
6
7
The expected
.176 .125 .097 .079 .067 .058
value is μx = 3.441.
8
.051
μx = 1(.301) + 2(.176) + 3(.125) + 4(.097) + 5(.079) +
6(.067) + 7(.058) + 8(.051) + 9(.046)
= 3.441
9
.046
Probability Histogram for
equally likely outcomes 1 to 9
In this uniform
distribution, the
mean 5 is
located at the
center.
Probability Histogram for
Benford’s Law
The mean is
3.441 in this
right skewed
distribution.
Recall…

Computing a measure of spread is an
important part of describing a distribution
(SOCS)

The variance and the standard deviation are
the measures of spread that accompany the
choice of the mean to measure center.
Variance of a Discrete
Random Variable

Suppose that X is a discrete random variable
We will use σx2 to
whose distribution is
signify the variance
Value of X: x1 x2 andxσ3x for…
the xk
Probability: p1 standard
p2 p3 deviation.
… pk

And that the mean μ is the mean of X. The
variance of X is
σx2 = (x1 – μx)2p1 + (x2 – μx)2p2 + … + (xk – μx)2pk

The standard deviation σx of X is the square root of the
variance.
Ex 3: Linda Sells Cars

Linda is a sales associate at a large auto
dealership. She motivates herself by using
probability estimates of her sales. For a
sunny Saturday in April, she estimates her
car sales as follows:
Cars Sold:
0
1
2
3
Probability:
0.3
0.4
0.2
0.1
Find the mean and variance.
xi
pi
xipi
(xi – μx)2pi
0
0.3
0.0
(0 – 1.1)2(0.3) = 0.363
The
0.004
standard
deviation is
σx = 0.943
1
0.4
0.4
(1 –
2
0.2
0.4
(2 – 1.1)2(0.2) = 0.162
3
0.1
0.3
(3 – 1.1)2(0.1) = 0.361
μx = 1.1
1.1)2(0.4)
=
σx2 = 0.890
The Law of Large Numbers



Draw independent observations at random
from any population with finite mean μ.
Decide how accurately you would like to
estimate μ.
As the number of observations drawn
increases, the mean x of the observed values
eventually approaches the mean μ of the
population as closely as you specified and
then stays that close.
Ex 4: Heights of Young Women
(Law of Large Numbers)
The average
height of young
women is 64.5
in.
The Law of Small Numbers




The law of small numbers does not exist,
although psychologists have found that most
people believe in the law of small numbers.
Most people believe that in the short run,
general rules of probability with be consistent.
This is a misconception because the general
rules of probability only exist over the long run.
In the short run, events can only be
characterized as random.
How large is a large number?



The law of large numbers does not state how
many trials are necessary to obtain a mean
outcome that is close to μ.
The number of trials depends on the
variability of the random outcomes.
The more variable the outcomes, the more
trials that are needed to ensure that the mean
outcome x is close the distribution mean μ.