EXPECTED VALUE AND STANDARD DEVIATION OF
RANDOM VARIABLES
E ( X ) = µ x = ∑ xi ⋅ p ( X = xi )
Expected Value of X
Variance of X
2
Var( X ) = σ x = ∑[(xi − µx ) 2 ⋅ p( X = xi )]
Standard Deviation of X
σ x = Var ( X )
Rules for Expected Value:
Linear Transformation
E (a ⋅ X + b) = a ⋅ E ( X ) + b
Combination
E ( X ± Y ) = E ( X ) ± E (Y )
Rules for Variance:
Linear Transformation
Var (a ⋅ X + b) = a 2 ⋅ Var ( X )
Combination
If X and Y are independent, then
Var ( X ± Y ) = Var ( X ) + Var (Y )
EXPECTED VALUE AND
STANDARD DEVIATION OF A
RANDOM VARIABLE
Using a fair six-sided
sided die, determine the expected value and standard deviation of a single roll.
From a probability distribution table we can observe the outcomes and the assigned probabilities
of each outcome.
X = xi
1
2
3
4
5
6
P(X = xi)
1/6 1/6 1/6 1/6 1/6 1/6
µ X = E ( X ) = ∑ x i ⋅ P ( X = x i ) =x1 ⋅ P ( X = x1 ) + x 2 ⋅ P ( X = x 2 ) + ... + x 6 ⋅ P ( X = x 6 )
We will use our TI-83+
83+ to assist with the calculation. Enter the outcomes into L1 and the
assigned probabilities into L2.
Or
1Var Stats on L1,L2
Explanation in context: The expected value of rolling a die is 3.5, where E(X) = 3.5.
3.5
Now for variance and standard deviation: Using the expected value of rolling a die of 3.5 as µx
we can compute the difference between the outcomes, x, and the population mean, µx.
Variance is denoted by Var(X) = σx2. Also, standard deviation is denoted by σx. Recall,
standard deviation of random variable X is equal to the square root of Variance of X:
σx =
σ x2 .
2
2
Var ( X ) = σ x = ∑ ( xi − µ ) ⋅ P ( X = xi )
Standard Deviation of X = σ X = Var ( X ) =
35
or 2.9166666667 = 1.7078
12
At the top of L3, enter (L1 - 3.5)2*L2.
This represents the product of assigned probabilities and the squared differences between
outcomes and expected values. Or look at σ X on 1Var Stats.
Explanation in context: The variance of rolling a die is σx2 = 2.917.. Therefore the standard
deviation of rolling a die is the square root of the variance, then σx = 1.708.
TIME TO MANIPULATE THE RANDOM VARIABLES
EXAMPLE 1: Let’s suppose we change the face values of the six sided fair die to {11, 12, 13,
14, 15, 16}. How will adding 10 to each face of die affect the expected value and variance?
Define Y = X + 10 = {11, 12, 13, 14, 15, 16}.
µY = E (Y ) = y1 ⋅ P(Y = y1 ) + y2 ⋅ P(Y = y2 ) + ... + y6 ⋅ P(Y = y6 )
E(Y) =
This is called a linear transformation
transformation.. The advantage to this knowledge is if you know the
Expected Value of a random variable (X) and you add a constant value to each of the x
outcomes, then the Expected Value of (Y) is simply the E(X) + 10. See formula.
Here is the linear transformation of Y = X + 10. Did the variance change when adding a constant
to each x outcome for Y?
2
2
Var (Y ) = σ Y = ∑ ( yi − µ ) ⋅ P (Y = yi )
EXAMPLE 2: What would happen to the expected value and variance if we multiplied each x
outcome by 10 assuming a six sided fair die is being used?
Define Z = {10, 20, 30, 40, 50, 60}
2
2
Var ( Z ) = σ Z = ∑ ( zi − µ ) ⋅ P (Z = zi )
EXAMPLE 3: Another linear transformation: So what would be the expected value (mean) and
variance, if we multiplied each face value of six sided fair die by ten and then added 5?
Define K = 10X + 5 = {15, 25, 35, 45, 55, 65}
Dot Plot
Collection 1
0
Collection 1
10
0
10
20
30
40
50
60
X
20
30
70
Dot Plot
40
50
60
70
Y
Dot Plot
Collection 1
0
10
20
30
40
50
60
70
z
Dot Plot
Collection 1
0
10
20
30
40
50
60
70
k
Let’s work together: From the data collected, we found the following information from 100
randomly selected students at CHS regarding the number of siblings a student had (this does not
include themselves).
# of siblings, x
0
1
2
3
4
5
Proportion, P(X = x)
.12
.35
.23
.16
.09
.05
What is the expected number of siblings for CHS students? E(X) =
________
What is the standard deviation of number of siblings? σx = _________
LINEAR TRANSFORMATIONS & COMBINATIONS
Cell phone companies have different plans allowing customers to have varying number of
minutes per plan. They know the total amount customers pay monthly for their basic service
charge of plans, but is looking at gauging customers more. The chart represents the number of
minutes customers go beyond their allotted time plan. For example, a plan allows for a total of
500 minutes per month but the customer used the phone for 560 minutes exceeding the plan by
60 minutes.
# of excess minutes,
0
15
30
45
60
75
90
105
x
P(X = x)
.37
.26
.17
.08
.05
.02
.02
.03
What is the expected number of excess minutes used? E(X) = ________
What is the variation of number of excess minutes? σx2 = _________
To increase the cell phone’s profit, the company places a $.012 roaming charge on excess
minutes beyond the plan allotment. What is the expected profit knowing the expected number of
excess minutes? Also, what is the standard deviation of profit knowing the expected number of
excess minutes? Let Y represent the profit based on excess minutes used on cell phones.
E (Y ) = a ⋅ E ( X ) + b
E (Y ) = .012 ⋅ E ( X ) + 0
E (Y ) = ____________
Var (Y ) = Var (a ⋅ X + b) = a 2 ⋅ Var ( X )
2
Var(Y ) = Var(.012 ⋅ X + 0) = (.012) ⋅ Var( X )
Var (Y ) = ___________
At Christmas, the cell phone company gives all cell phone owners a holiday gift of 10 additional
free minutes beyond their plan. What is the expected number of holiday excess minutes? What is
the standard deviation of the holiday excess minutes? Let H represent the number of excess
minutes used during the holiday month based on the expected number of excess minutes used
regularly.
E(H ) = a ⋅ E( X ) + b
E ( H ) = 1 ⋅ E ( X ) − 10
E (H ) = ___________
Var ( H ) = Var (1 ⋅ X − 10) = 12 ⋅ Var ( X )
Var(H ) = ___________
This is the grade distribution for the 2001 AP Statistics Exam at OHS.
AP Grade
1
2
3
4
5
Probability
0.00
0.11
0.35
0.33
0.21
The Statistics Department decides to rescale the AP Grade to determine what grade would have
been obtained if the test had been based on 100 points. The Statistics teacher works out a scaled
grade based on Grade Scale = 60 + 6.868(AP Grade) . What is the scaled average and standard
deviation of the scaled grade based on the expected AP Grade? The expected AP Grade is 3.64
and the variation is .8704.
E (Y ) = a ⋅ E ( X ) + b
E (Y ) = _____________
Var (Y ) = Var (a ⋅ X + b) = a 2 ⋅ Var ( X )
Var (Y ) = _____________
Assignment:
1. A commuter airline flies small planes between San Antonio and Corpus Christi. For small
planes, the baggage weight is a concern, especially on foggy mornings, because the weight of
the plane has an effect on how quickly the plane can ascend. Suppose that it is known that the
variable w = weight of baggage checked by a randomly selected passenger has a mean and
standard deviation of 42 and 16 respectively. Consider a flight on which ten passengers, all
traveling alone, are flying. If we use wi to denote the baggage weight for passenger i (for i
ranging from 1 to 10), the total weight of checked baggage, t, is then t = w1 + w2 + …+ w10.
Note that t is a linear combination of the wi’s.
The expected mean value of t is __________.
Since the passengers are traveling alone, it is reasonable to think the ten baggage weights are
unrelated and that the wi’s are independent. Then the variance of t is ____________.
2. The senior class is selling raffle tickets at the Senior Auction and Dinner night. Each raffle
ticket costs $10 and they plan on selling 200 raffle tickets. The prizes, which have been paid
for from the money collected from the sale, are one $100 stereo system, two $75 Dillard's
gift certificates, four $30 Chili's gift certificates, and eight $15 AMC gift certificates. What
is the expected value of a raffle ticket? How much profit per raffle ticket is the senior class
expecting to make?
3. A study of how well freshmen adapt the first six weeks of the school year, 50 randomly
selected 9th graders office referral records were examined to determine how often the ninth
graders needed administrative guidance to a positive road to follow on campus.
# of referrals, x
0
1
2
3
4
5
6
P(X = x)
.46
.22
.12
.10
.06
.02
.02
What is the expected number of referral for OHS 9th graders? E(X) = ________
What is the standard deviation of number of referrals? σx = _________
4. A nationwide standardized exam consists of a multiple-choice section and a free response
section. For each section, the mean and standard deviation are reported to be
Mean
Standard deviation
Multiple-Choice
38
6
Free Response
30
7
Let’s define x1 and x2 to be the multiple choice and free response score, respectively, of a
student selected at random from those taking the exam. We are also interested in the variable
y = total score. Suppose that the total score is computed as y = x1 + 2x2. What are the mean
(expected value) and standard deviation of y?
µ y = µ x + 2 x = µ x + 2 µ x = _________
1
2
1
2
But why can’t we calculate the standard deviation? What condition is required and how is it
violated? __________________________________________________________________
5. Natalie and Ryan are cross country runners for CHS. Each morning practice they
run 3 miles on average about 19 and 18 minutes, respectively. The standard deviation
of time is approximately 2 and 3 minutes, respectively. We will assume that they run
independently. What is their expected difference in time to run three miles? And what
is the standard deviation difference in time to run three miles? Hint: careful with
standard deviation!
6. Suppose the mean SAT verbal score is 525 with standard deviation 100, while the mean SAT
math score is 575 with standard deviation of 100. What can be said about the mean and
standard deviation of the combined math and verbal scores? Calculate each if possible.
7. Our band members decide to wrap Christmas presents for folks as a fundraiser. Freshman
band members collect present and donation for gift wrapping service with a mean time 4
minutes and standard deviation of 1 minute. The sophomores are responsible for wrapping
the present with an average time of 12 minutes and standard
deviation of 3 minutes. The juniors are in charge of tying ribbon
and placing bows on gifts with a mean time of 2 minutes and
standard deviation of 30 seconds. The seniors return wrapped gift
to folks with an average time of 3 minutes and standard deviation
of 1 minute. What is the mean and standard deviation of this
assembly line approach to fundraising?
8. Consider a small ferry that can accommodate cars and buses. The toll for
cars is $3 and the toll for buses is $10. Let x and y denote the number of
cars and buses, respectively, carried on a single trip. Suppose that x and y
are independent and have the probability distribution below.
X
0
1
2
3
4
5
P(X)
.05
.10
.25
.30
.20
.10
a)
b)
c)
d)
e)
f)
Y
0
1
2
P(Y)
.50
.30
.20
Compute the mean and variance of X.
Compute the mean and variance of Y.
Compute the mean and variance of the total amount of money collected in the tolls from
cars.
Compute the mean and variance of the total amount of money collected in the tolls from
buses.
Compute the mean and variance of W = total amount of money collected in tolls.
Compute the mean and variance of Z = total number of vehicles (cars and buses) on the
ferry.
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