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SPEED REVIEW
CHAPTER 7
 A random variable is a variable whose value is a numerical outcome of a random phenomenon.
 A discrete random variable is one in which there are a countable number of outcomes. The
distribution of a discrete random variable is a table (or histogram) showing each possible outcome
along with the probability of that outcome. To find the probability of a discrete random variable, add
the probabilities of all of the outcomes in the range.
 A continuous random variable is one in which the variable takes on every possible value in an
interval. The distribution of a continuous random variable is a density curve. To find the probability
for a continuous random variable, find the area under the density curve.
 A normal distribution is a special distribution of a continuous random variable.
 The mean of a random variable or expected value is (on formula sheet) E( X ) = Σxi pi . (Multiply
each outcome by its probability and add them all up.) This gives the average outcome per game if the
phenomenon were repeated many times.
 The variance of a random variable is (on formula sheet) VAR( X ) = Σ(xi −μx)2 pi ; to get standard
deviation, you square root the variance.
 You can find the mean and standard deviation of a random variable on your calculator by entering
outcomes in L1 and probabilities in L2 and creating L3 = L1 * L2. Then use 1-Var Stats for L3.
 Special rules:
a) If a constant is added to every number in a distribution, then the mean of the new distribution is
the old mean plus the constant and the standard deviation stays the same. (Adding does not change
the spread; it just shifts distribution.)
b) If every number in a distribution is multiplied by a constant, then the mean of the new distribution
is the old mean times the constant and standard deviation of the new distribution is the old
standard deviation times the constant. (The variance is the square of the standard deviation so the
variance is multiplied by the square of the constant.)
c) If a new distribution is formed by adding or subtracting randomly selected individuals from two
existing distributions then the mean of the sums (or differences) is the sum (or difference) of the
means. That is, xy x y or xy x y . This calculation works regardless of whether the
observations are independent.
d) If a new distribution is formed by adding or subtracting randomly selected individuals from two
existing distributions then the variance of the sums (or differences) is the sum of the variances,
provided that the observations are independent. That is, σ2X+Y = σ2X + σ2Y . To find the standard
deviation, square root this result. Note that we always ADD variances.
Extremely important for success on the AP Statistics Exam: If you don’t remember the
“special rules” AND you have time, create a small data set in L1, transform the data into L2,
then use 1-Var Stats to evaluate the effect the transformation has on mean, variance, and
standard deviation.
Practice
Practiceproblems
problems on the other side
1. Which of the following is a discrete random variable?
I. The number of cans of Coke on a beverage delivery truck.
II. The weight of a single can of Coke.
III. The number of cans of Coke in a 12-pack of Coke at Shop-Rite.
(A) I only
(B) II only
(C) III only
(D) I and III
(E) I, II and III
2. A fast food restaurant just leased a new freezer and food fryer for three years. The service
contract for the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service
charge for each repair needed. The restaurant’s research suggested that during a given
year 80% of these freezers need no repairs, 11% needed to be serviced once, 5% twice, 4$
three times, and none required more than three repairs
a. Find the expect number of repairs this kind of freezer is expected to need each year.
b. Find the standard deviation of the number of repairs each year.
c. What are the mean and standard deviation of the restaurant’s annual expense for the
service contract?
d. How many times should the restaurant expect to have to get this freezer repaired over
the three-year term of the lease?
e. What is the standard deviation of the number of repairs that may be required during the
three-year term of the lease? On what assumption does your calculation rest?
Solutions:
1. The correct answer is A. The number of cans of Coke on a truck is countable (an integer
value) and it results from a random process; so it is a discrete random variable. The weight
of a single can of coke may not be an integer, so it is not a discrete variable. And the number
of cans of Coke in a 12-pack is an integer, but it does not vary and it does not result from a
random process; so it is not a random variable.
2.
a. E(X) = 0(0.80) + 1(0.11) + 2(0.05) + 3(0.04) = 0.33 repairs.
b. Var(X) = (0–0.33)2(0.80) + (1–0.33)2(0.11) + (2–0.33)2(0.05) + (3–0.33)2(0.04) = 0.561
Standard deviation = 0.561 = 0.749
Tip: You can also put {0, 1, 2, 3) in L1 and {80, 11, 5, 4} in L2 and run 1-Var Stats L1, L2
c. Let C = $125 + $35X; E(C) = $125 + $35(0.33) = $136.55
Standard deviation (C) = $35(0.749) = $26.22
d. E(X1 + X2 + X3) = 0.33 + 0.33 + 0.33 = 0.99 repairs
e. Var(X1 + X2 + X3) = 0.561 + 0.561 + 0.561 = 1.683, so Std Dev (C) = 1.297
The assumption is that the number of repairs is independent from year to year. (This
might be incorrect because some freezers might need more service than others.)