Download expected value and standard deviation of random

Ch15 Linear Transformations
Name: ______________________
Expected Value of X
E( X )  x   xi  p( xi )
Variance of X
Var ( X )   x 2  [( xi  x )2  p( xi )]
Standard Deviation of X
 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 )
Example 1: Using a fair six-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
Example 2: 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}. Find the mean and standard deviation of Y.
This is called a linear 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.
Example 3: a) What would happen to the expected value and standard deviation if we multiplied each x outcome by 10 assuming a
six sided fair die is being used?
b) 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?
Example 4: 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?
Practice: Do your work on a separate sheet of paper.
1. A study of how well freshmen adapt the first six weeks of the school year was done. 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
a. What is the expected number of referrals?
b. What is the variance and standard deviation of number of referrals?
If we add 7 to the list of x above and then multiply it by -2 then:
c. What is the new mean number of referrals?
d. What is the new variance and standard deviation?
2. Suppose the equation Y = 200 + 10X converts a PSAT math score, X, into an SAT math score, Y. Suppose the average PSAT math
score is 48 with a standard deviation of 1.5.
a. What is the average SAT math score?
b. What is the variance and standard deviation for the SAT math score?
3. This is the grade distribution for the 2001 AP Statistics Exam at CHS.
AP Grade
1
2
3
4
5
Probability
0.00
0.02
0.06
0.44
0.48
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 standard deviation (variation) is .8704.
4. A random variable x has the following probability model:
With
x  0.4 and  x 2  1.04
a. If we multiply the list by 2 and then add 3 to every value find the new mean, variance, and standard deviation.
b. If we multiply the entire list by -5 find the new mean, variance, and standard deviation.
c. Let Y= -3X + 4. Find the new mean, variance, and standard deviation.
5. Toss four fair coins and let X equal the number of heads observed. Construct a probability distribution for X. Find P(X > 2).
6. Suppose that you have the choice of receiving $500 in cash or receiving a gold coin that has a face value of $100. The actual value
of the gold coin depends on its gold content. You are told that the coin has a 40% chance of being worth $400, a 30% chance of being
worth $900, and 30% chance of being worth its face value. If you base your decision on expected value, which should you choose?
7. A lottery that sells 150,000 tickets has the following prize structure:
First prize of $50,000, Five second prizes of $10,000, 25 third prizes of $1000, 1000 fourth prizes of $10
Compute the expected value of the ticket.
8. A club sells raffle tickets for $5 each. There are 10 prizes of $25 and one prize of $100. If 200 tickets are sold, and you bought one
of them, what are your expected winnings? Have you paid too much for the ticket?
9. A large university sponsors a raffle and sells 2400 tickets.
What is the expected value of a ticket if there is one prize worth $500, four prizes worth $100, and ten prizes worth $10? What is the
standard deviation?
10. Make sure you know how to find probabilities of discrete random variables and continuous random variables.