Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1)
A random variable X is exponentially distributed with a mean of 10. Determine a so that
Pr(0 x a) = 0.75.
1)
A)
a =1
10
B)
a =1
10 ln 4
C)
a = ln 3
4
D)
a = ln 0.75
E)
none of these
2)
Which of the graphs below could not possibly be the graph of a probability function f(x)?
2)
A)
graphs A and C
B)
graph B only
C)
graphs A and B
D)
graphs B and C
E)
none of these
3)
A table saw cuts construction studding. Observation has shown that the lengths of the studs are
normally distributed with a mean of 10 feet and a standard deviation of 6 inches. Which of the
following correctly represents the probability that a randomly chosen stud exceeds 11 feet?
3)
A)
1
0.5 2
11
e(–1/2)[(x – 10)/0.5]2 dx
B)
1
10.5
9.5
e–[(x – 10)/5]2 dx
C)
1
2
10.5
9.5
e–x2/2 dx
D)
1
2
11
e–x2/2 dx
E)
none of these
4)
Suppose X is a random variable whose probabilities are Poisson distributed with pn=(14)n
n! e–14.
Which of the following is true?
4)
A)
The expected value of X is e–14.
B)
The probability that x = 0 is zero.
C)
The standard deviation of X is 14.
D)
The probability that x = 14 is approximately 0.1060.
D
5)
A set of exam scores is 80, 75, 85, 90, 100, 70, 60. The standard deviation equals
5)
A)
10
B)
20
C)
50
D)
7
E)
none of these
E
A
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
6)
A car dealer records the number of Mercedes sold each week. During the past 50 weeks,
there were 15 weeks with no sales, 20 weeks with one sale, 10 weeks with two sales, and 5
weeks with three sales. Let X be the number of Mercedes sold in a week selected at
random from the past 50 weeks. Compute E(X). Enter just a real number rounded off to
one decimal place (no label).
6)
7)
A random variable X has a density function f(x) =1
ln 16 ·1
x , 1 x 16. Find a such that
Pr(1 X a) =3
4.
Enter just an integer, no labels.
7)
Missed work hours caused by one of a class of industrial accidents has a probability density function
f (t) =1
8e–t+3
8e–t/2 +1
24 e–t/3 where t is measured in hours.
8)
What proportion of these accidents result in 5 or fewer missed work hours?
Enter just a real number to two decimal places.
8)
9)
Determine the probability of an outcome of the probability density function
f(x) =12x2– 12x3being between 1
2 and 1 where 0 x 1.
Enter just a reduced fraction.
9)
10)
Dr. Smith‘s test score distribution is characterized by the probability density function
f(x) =x(10,000 –x2)
25,000,000 , 0 x 100. What percentage of people are likely to get a 60 or above
on the exam? Enter just a real number to two decimal places (no units).
10)
11)
A person throws a die until the side with two spots appears. The probability of throwing
the die exactly n times before throwing a “2” is 5
6
n1
6, n 0. What is the probability that
the number of throws before throwing a “2” is even? Enter just a reduced fraction.
11)
12)
A farmer has observed that the time to maturation of a certain crop is approximately
normally distributed with a mean of 60 days and a standard deviation of 2 days. Find the
percentage of plants that will mature in less than 55 days. Enter the percentage as just a
real number rounded off to two decimal places followed by %.
12)
13)
If f(x) =1
8x is a probability density function for 0 x 4, find F(x), the corresponding
cumulative distribution function and use it to find Pr(1 X 3).
Enter just a reduced fraction representing Pr(1 X 3). Do not label.
13)
14)
A new car dealer observes that the number X of warranty claims for repairs on each new
car sold is Poisson distributed, with an average of six claims per car. Compute the
probability that a new car sold by the dealer will have no more than three warranty claims.
Enter your answer in the form aeb.
14)
15)
The life of a battery is a random variable with probability density function f(x) =3
56 x2,
2 x 4 where x is the time in months. Calculate E(X).
Enter just a reduced fraction of form a
b unlabeled.
15)
4
16)
Consider a square with sides of length 2 as in the diagram below. An experiment consists
of choosing a point at random from the square and noting its x–coordinate. If X is the
x–coordinate of the point chosen, find the cumulative distribution function of X. [Recall
F(x) = Pr(0 X x).]
Enter just an unlabeled polynomial in x in standard form.
16)
17)
John would like to place a two dollar bet on his favorite racehorse, Black Velvet. He can
bet that Black Velvet will win or show (finish in the top three horses). If he bets correctly
that Black Velvet wins, he wins $20. If he bets correctly that Black Velvet shows, he wins
$7. John figures Black Velvet has a 20% chance of winning and a 70% chance of showing.
If X is the amount of money John wins if he bets Black Velvet will win and Y is the amount
of money he wins if Black Velvet will show, find E(X) and E(Y) . Enter just two real
numbers rounded off to two decimal places in the order given above representing dollars
(no units).
17)
18)
The probability density function for a random variable X is f(x) =2 ln x
(ln 4)2 x , 1 x 4. Find
Pr(1 X 2).
Enter just a reduced fraction.
18)
19)
Find (by inspection) the expected value and standard deviation of the random variable
with the following density function: f(x) =7
2e–49/2(x – 3.9)2
Enter your answer as just two numbers (a real number to one decimal place followed by a
reduced fraction) separated by a comma, the first representing E(X) and the second
representing Var(X).
19)
5
20)
Is f(x) =1
21 x2 a probability density function on the interval 1 x 4 ?
Enter “yes” or “no”.
20)
21)
The probability density function for a random variable X is f(x) =3
4(2x –x2), 0 x 2 .
Find Pr(0 X 1).
Enter just a reduced fraction.
21)
22)
It is estimated that the time between arrivals of visitors to a public library is an exponential
random variable with expected value of 13 minutes. Find the probability that 30 minutes
elapses without any arrivals. Enter your answer as just ea/b, where a
b is a reduced
fraction.
22)
23)
Suppose f(x) =kx–5 is a density function for a random variable x for x 2. Find the value
of k and find the corresponding cumulative distribution function.
Enter your answer exactly as a ± bxc.
23)
24)
The table below is the probability table for a random variable X. Find the standard
deviation of X.
Outcome –3–2–1 1 2 3
Probability 0.1 0.1 0.4 0.3 0.05 0.05
Enter just a real number rounded off to two decimal places.
24)
25)
Find the value of k that makes f(x) = 3e–kx a probability density function on the interval
x 0.
Enter just an integer.
25)
26)
Find the expected value and variance for the random variable whose probability density
function is f(x) = 2(x – 1), 1 x 2.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
26)
27)
Find the expected value and variance for the random variable whose probability density
function is f(x) =3x2, 0 x 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
27)
28)
Suppose that during a certain part of the day, the number X of automobiles that arrive
within any one minute at a tollgate is Poisson distributed, and Pr(X = k) =4ke–4
1 · 2 · … · k .
What is the average number of automobiles that arrive per minute?
Enter just an integer.
28)
29)
The probability density function for a random variable X is f(x) =3x–4, x 1. Find Pr(2 X).
Enter just a reduced fraction.
29)
30)
Find (by inspection) the expected value and standard deviation of the random variable
with the following density function: f(x) =1
2 2e–1/8x2
Enter your answer as just two numbers (integers or reduced fractions) separated by a
comma, the first representing E(X) and the second representing Var(X).
30)
31)
Find the expected value and variance for the random variable whose probability density
function isf(x) = 4x – 1, 1
2 x 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
31)
32)
An appliance comes with an unconditional money back guarantee for its first 6 months. It
has been found that the time before the appliance experiences some sort of malfunction is
an exponential random variable with mean 2 years. What percentage of appliances will
malfunction during the warranty period? Enter your answer as just a ±eb, where a is an
integer and b is a real number to two decimal places. (no units).
32)
33)
The table below is the probability table for a random variable X. Find E(X).
Outcome –3–2–1 1 2 3
Probability 0.1 0.1 0.4 0.3 0.05 0.05
Enter just a real number rounded off to two decimal places.
33)
34)
A Christmas tree grower anticipates a profit of $80,000 in a usual season. There is however
a 10% chance of pine bark beetle infestation in which case 70% of the trees are destroyed
and profit is reduced to $24,000. The grower can spray for beetles at the beginning of the
season at a cost of $7,000. Compute E(X). Enter just an integer rounded off to the nearest
thousand.
34)
35)
Is f(x) =3
2x – 1 a probability density function for 0 x 2?
Enter “yes” or “no”
35)
36)
Determine the probability of an outcome of the probability density function f(x) =4x3
being between 1
4 and 1
2where 0 x 1.
Enter just a reduced fraction.
36)
37)
A basketball player attempts successive free throws until he succeeds in making a basket.
Suppose the probability of success of each attempt is 0.7; thus, the probability of exactly n
failures before the first success is (0.3)n(0.7), n 0. What is the probability that the number
of failures before the first successful free throw is odd? Enter just a reduced fraction.
37)
8
38)
The riders of the New Town Elementary school bus consists of 5 five year olds, 3 six year
olds, 10 eight year olds, 1 nine year old, 4 eleven year olds and a twelve year old. A child
is selected at random and her age is noted. Let X be the outcome. Find E(X). Enter just a
reduced fraction of form a
b (no label).
38)
39)
A carnival game costs $2 to play. A player draws a ball at random from a sack containing
1 white ball, 2 blue balls, 3 red balls, and 4 yellow balls. The payoff for drawing a
particular color ball is as follows: white pays $5, blue pays $4, red pays $3 and yellow pays
nothing. If X is the amount of money a player wins. Calculate E(X). Enter just a real
number rounded off to two decimal places (no label).
39)
40)
The table below is the probability table for a random variable X. Find Var(X).
Outcome –1 0 1 2
Probability 3
7
1
7
1
7
2
7
Enter just a reduced fraction of form a
b.
40)
41)
Given the density function f(x) =3
64 x2, 0 x 4, determine the corresponding cumulative
distribution function.
Enter just an unlabeled polynomial in x in standard form.
41)
42)
A lumber yard cuts 2″ x 4″ lumber into 8 foot studs. It is observed that the actual lengths of
the studs are normally distributed with mean 8 feet and standard deviation 1 foot. What
proportion of the studs are longer than 8.25 feet? Enter just a real number rounded off to
two decimal places.
42)
43)
Find the value of k that makes f(x) = k x a probability density function on the interval
4 x 9.
Enter just a reduced fraction.
43)
44)
Suppose f(x) = k(x2+ 2x) is a probability density function for a continuous random
variable on the interval 0 x 3. Find the value of k and find the corresponding
cumulative distribution function.
Enter just an unlabeled polynomial in x in standard form.
44)
45)
A random variable X has a density function f(x) =24
x3, 3 x 6. Find b such that
Pr(X b) = 0.4.
Enter your answer exactly in the reduced form a b
c, unlabeled.
45)
46)
Find the value of k that makes f(x) =kx2 a probability density function on the interval
0 x 1.
Enter just an integer.
46)
47)
Suppose f(x) =1
x2 is a probability density function for x 1. Find Pr(2 X 10).
Enter just a reduced fraction.
47)
10
48)
The probability density function of a continuous random variable X is f(x) =3
2x –3
4x2,
0 x 2. Is this the graph of f(x) with the shaded area corresponding to Pr 1
2 x 3
2?
Enter “yes” or “no”.
48)
49)
Find (by inspection) the expected value and the variance of the random variables with the
following density function: f(x) =1
4 2e–(1/2)[(x – 1)/4]2, –< x <
Enter your answer as just two integers separated by a comma, the first representing E(X)
and the second representing Var(X).
49)
50)
Is f(x) =1
(x + 1)2 is a probability density function for x 0? If so, find P(X 2).
Enter either “no” or just a reduced fraction of form a
b.
50)
51)
When a road crew inspects a road that hasn‘t been worked on for several years, then the
distance between necessary repairs is an exponential random variable with a mean of 0.25
miles. What is the probability that the crew will find a mile long stretch of road that does
not need repairs? Enter your answer as just eb.
51)
52)
Find (by inspection) the expected value and the variance of the random variables with the
following density function: f(x) = 0.2e–0.2x, x 0
Enter your answer as just two integers separated by a comma, the first representing E(X)
and the second representing Var(X).
52)
53)
When mice are placed in a certain maze the amount of time it takes them to go through the
maze is approximately normally distributed with a mean of 25 minutes and a standard
deviation of 5 minutes. What is the probability that a mouse will complete the maze in
under 30 minutes? (Hint: find the normal density function first). Enter just a real number
rounded off to two decimal places.
53)
54)
The table below is the probability table for a random variable X. Find E(X), Var(X), and the
standard deviation of X.
Outcome 40 50 60 70 80
Probability 0.3 0.15 0.15 0.2 0.2
Enter just three real numbers all rounded off to two decimal places: a, b, c representing the
three quantities in the order requested above, separated by commas (no labels).
54)
55)
The table below is the probability table for a random variable X. Find E(X).
Outcome –1 0 1 2
Probability 3
7
1
7
1
7
2
7
Enter just a reduced fraction of form a
b.
55)
56)
Suppose the number of cars passing through a toll booth in a 10 minute interval is a
Poisson random variable. If the average number of cars is 23, give an expression for the
probability that n cars pass through the booth. Is pn=(23)n
n! e–23 correct?
Enter “yes” or “no”.
56)
57)
A student taking five courses keeps a record of the number of assignments due each day in
all her courses. Over the course of the 60–day semester she finds on 20 days no
assignments are due, on 15 days an assignment is due in one course, on 15 days an
assignment is due in two courses, on 9 days an assignments is due in three courses and
once during the semester she has an assignment due in 4 courses. If X is the number of
assignments due on a day selected at random from the semester, find E(X).
Is E(X) = 0 ·1
3+ 1 ·1
4+ 2 ·1
4+ 3 ·3
20 + 4 ·1
60 the correct answer?
Enter “yes” or “no”.
57)
Missed work hours caused by one of a class of industrial accidents has a probability density function
f (t) =1
8e–t+3
8e–t/2 +1
24 e–t/3 where t is measured in hours.
58)
What proportion of these accidents result in more than 9 missed work hours?
Enter just a real number to two decimal places.
58)
59)
The table below is the probability table for a random variable X. Find the standard
deviation of X.
Outcome –1 0 1 2
Probability 3
7
1
7
1
7
2
7
Enter just a real number rounded off to two decimal places.
59)
60)
In a certain office, the number of typewriters that break down during any given week is
Poisson distributed with = 2. What is the probability that more than three typewriters
break down during a week? Enter your answer in the form a ±b
ced where b
c is reduced.
60)
61)
Find the expected value and variance for the random variable whose probability density
function is f(x) = x, 0 x 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
61)
62)
Given the probability density function f(x) =1
3, determine the corresponding cumulative
distribution function where 12 x 15.
Enter an unlabeled polynomial in x in standard form.
62)
63)
Suppose that a bag holds 3 blue balls and one red ball. We pull a ball from the bag at
random, return it and then repeat the process. Suppose we continue pulling balls until the
blue ball is drawn and then we observe the number of consecutive red balls drawn. What
is the average number of red balls between occurrences of blue balls? Is
E(X) =
n = 1
n3n
4n + 1 =1
4
n = 1
n3
4
n= 3 correct? Enter “yes” or “no”.
63)
64)
A hardware store will cut lumber any length between 5 and 20 feet. Say X is the length of
lumber requested by a customer. Then X is a uniform random variable with probability
density function f(x) =1
15 . Find E(X) and Var(X).
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
64)
65)
A random variable X has a cumulative distribution function F(x) = 1 –1
(x + 1)2 for x 0.
Find Pr(1 X 4).
Enter just a reduced fraction.
65)
66)
Find (by inspection) the expected value and standard deviation of the random variable
with the following density function: f(x) =e–0.1x.
Enter your answer as just two integers separated by a comma, the first representing E(X)
and the second representing Var(X).
66)
67)
Joe has a lawn mowing job. If he completes the work he earns $40. But there is a 30%
chance it may rain, in which case he won’t finish the job. He can pay Jane $20 to help him
and ensure that he finishes the job. If X is the amount Joe will get if he does not get Jane to
help, calculate E(X) and thus decide whether Joe should hire Jane or not. (If it rains,
assume Joe will make no money and if Joe hires Jane assume they will be able to finish the
job before it rains. Enter your answer exactly as a,b where a is an integer representing E(X)
in dollars (no units), and b is either “yes” or “no” answering the question “should Joe hire
Jane?”, separated by a comma.
67)
68)
A survey shows that the time spent in a checkout line in a certain supermarket has an
exponential density function with mean 5 minutes. What is the probability of spending 10
minutes or more in a checkout line? Enter just a real number rounded off to two decimal
places.
68)
69)
A random variable X has a cumulative distribution function F(x) =x
5– 2, 10 x 15. Find a
such that Pr(a X 15) =2
3.
Enter just a reduced fraction of form a
b.
69)
70)
Find the expected value and variance for the random variable whose probability density
function is f(x) =1
3,2 x 5.
Enter just two reduced fractions of form a
b (unlabeled) in the order E(X), Var(X) separated
by a comma.
70)
71)
Find the value of k that makes f(x) =kx3 a probability density function on the interval
0 x 1.
Enter just an integer.
71)
72)
Suppose a small amount of blood is sampled and the number of white blood cells are
counted. If the number of white blood cells is Poisson distributed with = 6, what is the
probability that the sample has more than 4 white blood cells? What is the average
number of white blood cells per sample? Is Pr(4 X) = 0.7149; E(X) = 6 correct?
Enter “yes” or “no”.
72)
73)
A random variable X has a probability density function f(x) =x
32 , 0 x 8. Find a such that
Pr(X a) =1
4.
Enter your answer exactly in the reduced form b c, unlabeled.
73)
74)
Find the expected value and variance for the random variable whose probability density
function is f(x) = 12x2(1 – x), 0 x 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a
comma.
74)
75)
Find the expected value and variance for the random variable whose probability density
function is f(x) =ex,x 0.(You may use the fact that lim
b beb= 0 .)
Enter just two integers (unlabeled) in the order E(X), Var(X) separated by a comma.
75)
76)
If f(x) = 6x(1 – x) is a probability density function for 0 x 1, find F(x), the corresponding
cumulative distribution function and use it to find Pr 1
2 X 1 .
Enter just a reduced fraction representing Pr 1
2 X 1 . Do not label.
76)
77)
The table below is the probability table for a random variable X. Find Var(X).
Outcome –3–2–1 1 2 3
Probability 0.1 0.1 0.4 0.3 0.05 0.05
Enter just a real number rounded off to two decimal places.
77)
78)
Let X be the time to failure of an electronic component, and suppose X is an exponential
random variable with E(X) = 4 years. Find the median lifetime, i.e., find M such that
Pr(X M) =1
2. Enter just a real number rounded to two decimal places (no units).
78)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
79)
Let X be a continuous random variable with a cumulative distribution function
F(x) = 1 –e–x2 (x 0). Find Pr(1 X 2).
79)
A)
e–1–e–2– 2
B)
e–1–e–2
C)
1 –e–1–e–2
D)
e–1–e–4
E)
none of these
80)
A random variable X has probability density function f(x) =ke–kx (x 1) for some constant k.
Suppose that Pr(1 X 2) =1
4, what is the value of k?
80)
A)
1
2 ln 2
B)
ln 2
C)
1
4
D)
3
2 ln 2
E)
none of these