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AP Statistics
Combining and Transforming Random Variables
1. Suppose X and Y are random variables with  X =35,  X =8,  Y =72,  Y =4. Given that X and Y are
independent variables, calculate the following:
 X Y 
 2 X Y 
 X Y 
 X Y 
 X Y 
 2X 
2. For a given high school basketball team, the number of baskets (X) for the leading scorer is E(X) =
8.3 with (X) = 1.25 and the number of baskets (Y) for the second leading scorer is E(Y) = 6.6 with
(Y) = 2.31.
a) Together, how many baskets would we expect each game?
b) What is the standard deviation of this number?
c) How many more baskets would we expect the leading scorer to have?
d) What is the standard deviation of this number?
e) Hypothetically, let’s say the leading scorer shoots only 3-point baskets and the second leading
scorer shoots 2-point baskets, how many points would we expect each game from the two
players?
f) What is the standard deviation of this number?
g) How many more points would we expect the leading scorer to have?
h) What is the standard deviation of this number?
3. The following are the distributions for the number of goals scored per soccer game for the three top
forwards from Vareebull High School in Random, Texas.
X
Xavier
0
1
2
P( X ) .10 .55 .35
Y
Yousuf
0
1
2
P(Y ) .20 .75 .05
Zoué
0
1
Z
2
3
P( Z ) .80 .10 .05 .05
a)
b)
c)
d)
e)
f)
Calculate the mean, variance and standard deviation for the number of goals that Xavier scores.
Calculate the mean, variance and standard deviation for the number of goals that Yousuf scores.
Calculate the mean, variance and standard deviation for the number of goals that Zoué scores.
Who is likely to win the scoring title for the team? Explain.
Who is the most consistent player? Explain.
Calculate the mean and standard deviation for the number of goals that all three players score
together.
g) Calculate the mean and standard deviation for the difference in goals between Xavier and Zoué.
4. Suppose X and Y are random variables with  X =100,  X =10,  Y =72,  Y =18. Given that X and Y
are independent variables, calculate the following:
 X2 Y =
 X Y =
 X Y =
 X2 =
 Y2 =
 X2 Y =
 X Y =
 X Y =
5. Chen, Hope, Marco and Panchali love to visit the local bakery that specializes in muffins. Listed
below are mean and standard deviation of the number of muffins each buys per visit.
Chen:  = 2.3 muffins,  = 1.1 muffins
Hope:  = 3.9 muffins,  = 2.2 muffins
Marco:  = 5.2 muffins,  = 0.7 muffins
Panchali:  = 4.2 muffins,  = 1.4 muffins
a) If all four friends visit the bakery, what is the expected number of muffins purchased? The
standard deviation?
b) How many more muffins would we expect Marco to purchase than Chen? The standard
deviation?
c) How many more muffins would we expect the males to buy than the females? The standard
deviation?
d) Hope promised to buy 4 extra muffins for her colleagues. How many muffins would we
expect her to purchase? The standard deviation?
e) The two ladies love going Thursday mornings because all muffins are $2.50 each. How much
would we expect them to spend? The standard deviation?
f) Marco buys the $4 mega muffins and Panchali buys the regular $3.25 muffins. How much
higher will Marco’s cost than Panchali’s? The standard deviation?
6. The probability distribution below represents (Y) the length of long distance calls in minutes.
Y
5 10 15 20 25
P(Y ) .1 .2 .3 .3 .1
a) What is the expected length of a long distance call? The variance and standard deviation?
b) Suppose that there is an initial connection charge of $.60 and a further charge of $.015 for each
minute. What is the expected cost of a long distance call? What is the standard deviation of
the cost of the call?
7. Some Japanese consumers are willing to pay premium prices for Maine lobsters. One factor in the
high cost is the mortality rate during the long-distance shipping from Maine to Japan. Suppose the
number of deaths X per crate of 10 lobsters is given by the following probability distribution.
X
0
1
2
3
4
5
P( X ) 0.48 0.28 0.17 0.05 0.01 0.01
a. How many living lobsters would we expect to arrive in Japan per crate?
b. If one restaurant wants only the lobster claws, how many claws would the shipping company
expect to lose from 3 crates?
8. The random variable X denotes the number of Granny Smith apples per purchase by supermarket
shoppers in the express lane. The probability distribution is given below:
X
1
2
3
P( X ) 0.35 0.40 0.25
a. Find the expected value, variance and standard deviation for the number of apples.
b. If the apples are 50¢ each, what is the expected cost? The standard deviation of cost?
c. A shopper in the express lane is expected to spend 72¢ on bananas with a standard deviation of
10¢. How much more would we expect the shopper to spend on apples than bananas? The
standard deviation?