Deck 3: Probability Distributions

Question 1

Experimental outcomes must occur as numerical values in order to define their probability distribution.

Answer 1

False

Question 2

In order to measure the dispersion of a random variable,look at its

A)standard deviation.
B)mean.
C)expected value.
D)average.

Answer 2

A

Question 3

The probability of a continuous variable having a specific value is 0.

Answer 3

True

Question 4

A numerical description of the outcome of an experiment is

A)a normal variable.
B)a discrete variable.
C)a random variable.
D)an experimental variable.

Question 5

A probability density function

A)gives the probability that the random variable equals a specific value of x.
B)is used for discrete random variables.
C)describes the graph whose underlying area represents probability over an interval.
D)each of the above is true.

Question 6

The uniform distribution defined over the interval from 25 to 40 has the probability density function

A)f(x)= 1/40 for all x
B)f(x)= 5/8 for 25 $\le$ x $\le$ 40 and f(x)= 0 elsewhere
C)f(x)= 1/25 for 0 $\le$ x $\le$ 25 and f(x)= 1/40 for 26 $\le$ x $\le$ 40
D)f(x)= 1/15 for 25 $\le$ x $\le$ 40 and f(x)= 0 elsewhere

Question 7

If x is normally distributed with mean 12 and standard deviation 2,then P(x $\le$ 9)is

A)P(z $\le$ 9/10).
B)P(z $\le$ -3/2)
C)P(z $\le$ 2/3)
D)P(z $\le$ -3/4)

Question 8

Larger values of the standard deviation result in a normal curve that is

A)shifted to the right
B)shifted to the left
C)narrower and more peaked
D)wider and flatter

Question 9

An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment.The random variable in this experiment is speed,measured in miles per hour.This random variable is a

A)discrete random variable
B)continuous random variable
C)complex random variable
D)None of these alternatives is correct.

Question 10

If one wanted to find the probability of ten customer arrivals in an hour at a service station,one would generally use the Poisson distribution.

Question 11

When the exponential distribution probability is given as P(x $\le$ 9 )= 1 - e^-9/18 the average value for x is

A)18
B)9
C)9/18
D)1/2

Question 12

Z is a standard normal random variable.The P(z > 2.11)equals

A)0.4821
B)0.9821
C)0.5
D)0.0174

Question 13

The expected value of the discrete random variable x is $\mu$ ,a weighted average of all possible values of x.

Question 14

The weight of an object,measured to the nearest gram,is an example of

A)a continuous random variable
B)a discrete random variable
C)either a continuous or a discrete random variable,depending on the weight of the object
D)either a continuous or a discrete random variable depending on the units of measurement

Question 15

A standard normal distribution is a normal distribution with

A)a mean of 1 and a standard deviation of 0
B)a mean of 0 and a standard deviation of 1
C)any mean and a standard deviation of 1
D)any mean and any standard deviation

Question 16

In a Poisson probability problem,the rate of errors is one every two hours.To find the probability of three defects in four hours,

A) $\lambda$ = 1,x = 4
B) $\lambda$ = 2,x = 3
C) $\lambda$ = 3,x = 2
D) $\lambda$ = 3,x = 6

Question 17

A random variable that can take on any value along a line segment is a continuous random variable.

Question 18

A statement that matches the values of a random variable with the probabilities of those values is

A)the expected value.
B)the variation of the random variable.
C)an experiment.
D)a probability distribution.

Question 19

Which of the following are continuous random variables? I.the weight of an elephant
II)the time to answer a questionnaire
III)the number of floors in a skyscraper
IV)the square feet of countertop in a kitchen

A)I and II only
B)III and IV only
C)I,II and IV
D)I,II,II,and IV

Question 20

Experiments with repeated independent trials will be described by the binomial distribution if

A)the trials are continuous.
B)each trial result influences the next.
C)the time between trials is constant.
D)each trial has exactly two outcomes whose probabilities do not change.

Question 21

A random variable that may assume only a finite or an infinite sequence (e.g. ,1,2,3,…)of values is a discrete random variable.

Question 22

The standard normal distribution is a normal distribution with a mean of 1 and a standard deviation of 1/3.

Question 23

The binomial probability distribution is most symmetric when p equals .5.

Question 24

The binomial distribution is appropriate to use to find the probability of the elapsed time between successes.

Question 25

Delicious Candy markets a two pound box of assorted chocolates.Because of imperfections in the candy making equipment,the actual weight of the chocolate has a continuous uniform distribution ranging from 31.8 to 32.6 ounces.
a.Define a probability density function for the weight of the box of chocolate.
b.What is the probability that a box weighs (1)exactly 32 ounces; (2)more than 32.3 ounces; (3)less than 31.8 ounces?
c.The government requires that at least 60% of all products sold weigh at least as much as the stated weight.Is Delicious violating government regulations?

Question 26

During lunch time,customers arrive at Bob's Drugs according to a Poisson distribution with $\lambda$ = 4 per minute.
a.During a one-minute interval,determine the following probabilities: (1)no arrivals; (2)one arrival; (3)two arrivals;and, (4)three or more arrivals.
b.What is the probability of two arrivals in a two-minute period?
c.What is the probability that no more than 30 seconds elapses between arrivals?

Question 27

Whenever the probability is proportional to the length of the interval,the random variable is uniformly distributed.

Question 28

The number of customers at Winkies Donuts between 8:00a.m.and 9:00a.m.is believed to follow a Poisson distribution with a mean of 2 customers per minute.
a.During a randomly selected one minute interval during this time period,what is the probability of 6 customers arriving to Winkies?
b.What is the probability that at least 2 minutes elapse between customer arrivals?

Question 29

An exponential distribution,like the Poisson distribution,can be described by a single parameter.

Question 30

The Poisson probability of three events in one hour is the same as the probability of 36 events in one day.

Question 31

For an exponentially distributed random variable x with a mean of 8,the probability of x < 8 is .5.

Question 32

A light bulb manufacturer claims his light bulbs will last 500 hours on the average.The lifetime of a light bulb is assumed to follow an exponential distribution.
a.What is the probability that the light bulb will have to replaced within 500 hours?
b.What is the probability that the light bulb will last more than 1000 hours?
c.What is the probability that the light bulb will last between 200 and 800 hours?

Question 33

If the Poisson distribution describes the number of occurrences per interval,the exponential distribution describes the length of the interval between occurrences.

Question 34

Weight,time,and temperature are examples of discrete random variables.

Question 35

The Poisson probability distribution is used with a continuous random variable.

Question 36

Suppose that the arrival time for the next bus at a bus stop is uniformly distributed between 1 and 25 minutes.The probability that the bus will arrive within the next 5 minutes is 1/6.

Question 37

Probabilities for discrete random variables are defined over intervals by a probability density function.

Question 38

The high school GPAs of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories.

a.Complete the table to make this a valid probability distribution.
b.What is the probability an applicant's GPA will be below 3.0?
c.What is the probability an applicant's GPA will be 2.5 or above?

Question 39

The probability density function matches each value of a continuous random variable with its probability.

Question 40

A random variable that has a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a binomial distribution.

Question 41

Assume that the time required to download a file from the Internet is exponentially distributed with mean equal to 4 minutes.What is the probability that a download will require at least 2 but not more than 4 minutes?

Question 42

The weight of a .5 cubic yard bag of landscape mulch in uniformly distributed over the interval from 38.5 to 41.5 pounds.
a.Give a mathematical expression for the probability density function.
b.What is the probability that a bag will weigh more than 40 pounds?
c.What is the probability that a bag will weigh less than 39 pounds?
d.What is the probability that a bag will weigh between 39 and 40 pounds?

Question 43

Consider a Poisson probability distribution in a process with an average of 3 flaws every 100 feet.Find the probability of
a.no flaws in 100 feet
b.2 flaws in 100 feet
c.1 flaws in 150 feet
d.3 or 4 flaws in 150 feet

Question 44

For a binomial distribution,compare P(x = 3)when n = 10 and p = .4 to P(x = 7)when n = 10 and p = .6.

Question 45

The time it takes to travel from home to the office is normally distributed with $\lambda$ = 25 minutes and $\sigma$ = 5 minutes.
a.What is the probability the trip takes more than 20 minutes?
b.What is the probability the trip takes less than 15 minutes.
c.What is the probability the trip takes between 30 and 35 minutes?
d.What is the probability the trip takes more than 40 minutes?

Question 46

Scores on an endurance test for cardiac patients are normally distributed with $\mu$ = 182 and $\sigma$ = 24.
a.What is the probability a patient will score above 190?
b.What percentage of patients score below 170?
c.What score does a patient at the 75th percentile receive?

Question 47

After a severe winter,potholes develop in a state highway at the rate of 5.2 per mile.Thirty-five miles of this highway pass through Washington County.
a.How many potholes would you expect to see in the county?
b.What is the probability of finding 8 potholes in 1 mile of highway?

Question 48

A video rental store has two video cameras available for customers to rent.Historically,demand for cameras has followed this distribution.The revenue per rental is $40.If a customer wants a camera and none is available,the store gives a $15 coupon for tape rental.

a.What is the expected demand?
b.What is the expected revenue?
c.What is the expected cost?
d.What is the expected profit?

Question 49

A manufacturer of computer disks has a historical defective rate of .001.What is the probability that in a batch of 1000 disks,2 would be defective?

Question 50

A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained.Following is a record of the number of students in a class of 20 who took the test each number of times.

a.Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition.
b.Find the expected value of the number of tests taken.
c.Compute the variance.
d.Compute the standard deviation.

Question 51

A process follows the binomial distribution with n = 8 and p = .3.Find
a.P(x = 3)
b.P(x > 6)
c.P(x $\le$ 2)

Question 52

Customers at a popular restaurant that refuses reservations arrive according to the Poisson distribution at a rate of 4 parties every 5 minutes.What is the probability that there will be more than 2 minutes between arriving parties?