Deck 12: Probability and Calculus

Question 1

The table below is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X.

A) E(X) = -0.2; Var(X) = 2.22; standard deviation of X =1.49
B) E(X) = -0.2; Var(X) = 2.16; standard deviation of X =1.47
C) E(X) = -0.2; Var(X) = 4; standard deviation of X =2
D) E(X) = -0.2; Var(X) = 0; standard deviation of X = 0
E) none of these

Answer 1

E(X) = -0.2; Var(X) = 2.16; standard deviation of X =1.47

Question 2

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).

Answer 2

0.20

Question 3

The table below is the probability table for a random variable X. Find Var(X). Enter just a reduced fraction of form .

Answer 3

Question 4

Is f(x) = a probability density function on the interval 1 ≤ x ≤ 4 ?

Question 5

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).

Question 6

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.

Question 7

The table below is the probability table for a random variable X. Find Var(X). Enter just a real number rounded off to two decimal places.

Question 8

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.

Question 9

The table below is the probability table for a random variable X. Find E(X). Enter just a real number rounded off to two decimal places.

Question 10

Which of the graphs below could not possibly be the graph of a probability function f(x)?

A) graphs B and C
B) graphs A and B
C) graphs A and C
D) graph B only
E) none of these

Question 11

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 (no label).

Question 12

Is f(x) = x - 1 a probability density function for 0 ≤ x ≤ 2?

Question 13

The table below is the probability table for a random variable X. Find the standard deviation of X. Enter just a real number rounded off to two decimal places.

Question 14

A set of exam scores is 80, 75, 85, 90, 100, 70, 60. The standard deviation equals

A) 7
B)
C)
D)
E) none of these

Question 15

The table below is the probability table for a random variable X. Find the standard deviation of X. Enter just a real number rounded off to two decimal places.

Question 16

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).

Question 17

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 the correct answer?

Question 18

Is f(x) = is a probability density function for x ≥ 0? If so, find P(X ≥ 2).
Enter either "no" or just a reduced fraction of form .

Question 19

The table below is the probability table for a random variable X. Find E(X). Enter just a reduced fraction of form .

Question 20

The table below is the probability table for a random variable X. Find E(X), Var(X), and the standard deviation of X. 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).

Question 21

Suppose f(x) = is a probability density function for x ≥ 1. Find Pr(2 ≤ X ≤ 10).
Enter just a reduced fraction.

Question 22

Determine the probability of an outcome of the probability density function being between where .
Enter just a reduced fraction.

Question 23

Let X be a continuous random variable with a cumulative distribution function . Find

A) -
B) -
C) 1 - -
D) - - 2
E) none of these

Question 24

The probability density function for a random variable X is f(x) = (2x - ), . Find Enter just a reduced fraction.

Question 25

The probability density function for a random variable X is f(x) = , x ≥ 1. Find .
Enter just a reduced fraction.

Question 26

A random variable X has a density function f(x) = , 3 ≤ x ≤ 6. Find b such that .
Enter your answer exactly in the reduced form a , unlabeled.

Question 27

The probability density function of a continuous random variable X is , . Is this the graph of f(x) with the shaded area corresponding to

Question 28

A random variable X has a density function f(x) = ∙ , 1 ≤ x ≤ 16. Find a such that .
Enter just an integer, no labels.

Question 29

Find the value of k that makes f(x) = 3 a probability density function on the interval Enter just an integer.

Question 30

A random variable X has probability density function f(x) = (x ≥ 1) for some constant k. Suppose that , what is the value of k?

A) ln 2
B) ln 2
C)
D) ln 2
E) none of these

Question 31

A random variable X has a cumulative distribution function F(x) = 1 - for . Find .
Enter just a reduced fraction.

Question 32

Find the value of k that makes f(x) = k a probability density function on the interval Enter just a reduced fraction.

Question 33

Find the value of k that makes f(x) = a probability density function on the interval .
Enter just an integer.

Question 34

A random variable X has a cumulative distribution function F(x) = 1 - (x≥1). Find Pr(a ≤ X ≤ 5).

A) 1 -
B) -
C) -
D)
E) none of these

Question 35

Determine the probability of an outcome of the probability density function being between where .
Enter just a reduced fraction.

Question 36

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 .
Enter just a reduced fraction representing Do not label.

Question 37

The probability density function for a random variable X is , Find Enter just a reduced fraction.

Question 38

Find the value of k that makes f(x) = kx a probability function on the interval .

A)
B)
C)
D)
E) none of these

Question 39

If f(x) = x 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 Do not label.

Question 40

Find the value of k that makes f(x) = a probability density function on the interval Enter just an integer.

Question 41

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.
Pr(a ≤ X ≤ b) =

Question 42

Find the expected value and variance for the random variable whose probability density function is Enter just two reduced fractions of form (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 43

Find the expected value and variance for the random variable whose probability density function is 0 ≤ x ≤ 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 44

A random variable has probability density function f(x) = 30 (1 - x
(0 ≤ x ≤ 1). Compute its cumulative distribution F(x).

A) 10 - + 6 + 1
B) 10 - 15 + 6
C) 30x(1 - x)
D) 30 - 60 +
E) none of these

Question 45

Suppose f(x) = 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 ± b .

Question 46

Suppose f(x) = k( + 2x) is a probability density function for a continuous random variable on the interval Find the value of k and find the corresponding cumulative distribution function.
Enter just an unlabeled polynomial in x in standard form.

Question 47

Find the expected value of the random variable whose density function is .

A)
B)
C)
D)
E) none of these

Question 48

Let X be a continuous random variable A ? X ? B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.

Pr(A ≤ X ≤ b) = F(b)

Question 49

Missed work hours caused by one of a class of industrial accidents has a probability density function where t is measured in hours.
What proportion of these accidents result in more than 9 missed work hours?
Enter just a real number to two decimal places.

Question 50

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.
(x) = f(x)

Question 51

Let X be a continuous random variable A ? X ? B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.

Question 52

A random variable X has a cumulative distribution function F(x) = - 2, . Find a such that Enter just a reduced fraction of form .

Question 53

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.

Question 54

A random variable X has a probability density function f(x) = , . Find a such that Enter your answer exactly in the reduced form b , unlabeled.

Question 55

Let X be a continuous random variable A ≤ X ≤ B and let f (x) be its probability density function and F (x) its cumulative distribution function. Indicate whether the following statements are true or false.
f(A) = 0, f(B) = 1

Question 56

Missed work hours caused by one of a class of industrial accidents has a probability density function where t is measured in hours.
Dr. Smith's test score distribution is characterized by the probability density function , . 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).

Question 57

Given the probability density function f(x) = , determine the corresponding cumulative distribution function where 12 ≤ x ≤ 15.
Enter an unlabeled polynomial in x in standard form.

Question 58

Given the density function f(x) = , 0 ≤ x ≤ 4, determine the corresponding cumulative distribution function.
Enter just an unlabeled polynomial in x in standard form.

Question 59

Find the expected value and variance for the random variable whose probability density function is Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 60

Missed work hours caused by one of a class of industrial accidents has a probability density function where t is measured in hours.
What proportion of these accidents result in 5 or fewer missed work hours?
Enter just a real number to two decimal places.

Question 61

Find (by inspection) the expected value and the variance of the random variables with the following density function: Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing Var(X).

Question 62

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: Enter your answer as just two numbers (integers or reduced fractions) separated by a comma, the first representing E(X) and the second representing .

Question 63

Find (by inspection) the expected value and the variance of the random variables with the following density function: f(x) = , -∞ < x < ∞
Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing Var(X).

Question 64

The life of a battery is a random variable with probability density function , where x is the time in months. Calculate E(X).
Enter just a reduced fraction of form unlabeled.

Question 65

Find the expected value and variance for the random variable whose probability density function is ≤ x ≤ 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 66

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 , where is a reduced fraction.

Question 67

Find the expected value and variance for the random variable whose probability density function is (You may use the fact that = 0 .)Enter just two integers (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 68

Let X be the time to failure of an electronic component, and suppose X is an exponential random variable with . Find the median lifetime, i.e., find M such that . Enter just a real number rounded to two decimal places (no units).

Question 69

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: 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 .

Question 70

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 Find E(X) and Var(X).
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 71

Find the expected value and variance for the random variable whose probability density function is 0 ≤ x ≤ 1.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 72

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 ± , where a is an integer and b is a real number to two decimal places. (no units).

Question 73

Find (by inspection) the expected value and standard deviation of the random variable with the following density function: .
Enter your answer as just two integers separated by a comma, the first representing E(X) and the second representing .

Question 74

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.

Question 75

Suppose X is a normal random variable with density function . Find the expected value and standard deviation of X.

A) μ = -4, σ = 1
B) μ = 1, σ = 4
C) μ = 1, σ = 16
D) μ = 4, σ = 1
E) μ = -4, σ = 16

Question 76

A random variable X is exponentially distributed with a mean of 2. Find

A) ( - )
B)
C)
D) -
E) none of these

Question 77

Find the expected value and variance for the random variable whose probability density function is 1 ≤ x ≤ 2.
Enter just two reduced fractions (unlabeled) in the order E(X), Var(X) separated by a comma.

Question 78

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?

A)
B)
C)
D)
E) none of these

Question 79

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 .

Question 80

A random variable X is exponentially distributed with a mean of 10. Determine a so that

A) a = ln 0.75
B) a =
C) a = ln 4
D) a = ln
E) none of these