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MEAN & VARIANCE OF
RANDOM VARIABLES
Chapter 7
EXPECTED VALUE
The mean (center) of a discrete random
variable, X.
It is a weighted average
-Each value of X is weighted by its
probability.
FINDING EXPECTED VALUE
To Find:
 x  x1 p1  x2 p2  ...xk pk   xi pi
Or in other words:
Multiply each value of X by its probability and
add products
This terminology is often used in relation to game of
chance or business prediction.
USING EXPECTED VALUE
Example:
Matthew is a doorman. The following table
gives the probabilities that customers will give
various amounts of money:
Amounts of
$
30
35
40
45
50
55
60
Probability
.45
.25
.12
.08
.05
.03
.02
What size tip can Matthew expect to get?
EXAMPLE 2
Concessionaires know that attendance at a
football stadium will be 60,000 on a clear day,
45,000 if there is light snow and 1500 if there is
heavy snow. Furthermore the probabilities of
clear skies, light snow or heavy snow on a
particular day is 1/2, 1/3, and 1/6 respectively.
What average attendance should be expected
for the game?
EXAMPLE 3
10,000 raffle tickets are sold at $2 each
for four prizes of $500, $1000, $1500,
and $5000. If you buy one ticket what is
the expected value of your gain?
LAW OF LARGE NUMBERS
As the number of observations increases,
the mean of the observed values, x ,
approaches the mean of the population, µ.
The more variation in the outcomes, the
more trials are needed to ensure that x is
close to µ .
RULES FOR MEANS
If X is a random variable and a and b are
fixed numbers, then
abx  a  bx
If X and Y are random variables, then
 X Y   X  Y
We use these for conversions!!!
EXAMPLE
The equation Y = 20 + 10X converts a
PSAT math score, X, into an SAT math score,
Y. The average PSAT math score is 48.
What is the average SAT math score?
EXAMPLE
Let  X  625 represent the average
SAT math score. Let Y  590
represent the average SAT verbal
score. What is the average combined
score?
VARIANCE OF A
DISCRETE RANDOM VARIABLE
If X is a discrete random variable with
mean , then the variance of X is
  x1   X  p1  x2   X  p2  ...  x k   X  pk
2
X
2
2
2
  x i   X 2 pi
The standard deviation  X  is the square
root of the variance.
RULES FOR VARIANCE
If X is a random variable and a and b are
fixed numbers, then

2
a  bX
b
2
2
X
If X and Y are independent random
variables, then 2
2
2
 X Y   X   Y

2
X Y
  
2
X
2
Y
EXAMPLE
The equation Y = 20 + 10X converts a
PSAT math score, X, into an SAT math score,
Y. The standard deviation for the PSAT
math score is 1.5 points. What is the
standard deviation for the SAT math score?
EXAMPLE
The standard deviation for the SAT
math score is 150 points, and the
standard deviation for the SAT verbal
score is 165 points. What is the
standard deviation for the combined
SAT score?
EXAMPLE
X and Y are independent random
variables. If  x  2.3 and  y  4.1,
find  x y .
ONE MORE…
As head of inventory for Knowway computer company, you
were thrilled that you had managed to ship 2 computers to
your biggest client the day the order arrived. You are
horrified, though, to find out that someone restocked
refurbished computers in with the new computers in your
storeroom. The shipped computers were selected randomly
from 15 computers is stock, but 4 of those were actually
refurbished.
If your client gets 2 new computers, things are fine. If
the client gets a refurbished computer, it will be sent back at
your expense - $100- and you can replace it. However, if
both computers are refurbished, the client will cancel the
order for this month and you will lose $1000. What is the
expected value and the standard deviation of your loss?