Transcript
Chapter 6: Discrete Distributions
Chapter Objectives
When you finish this chapter you should be able to
define a discrete random variable and a probability distribution.
solve problems by using the concepts of expected value and variance.
explain common discrete probability models and their parameters.
recognize the appropriate discrete model to use from the problem context.
find event probabilities for discrete models by using Excel, formulas or tables.
Quiz Yourself
True/False Questions
A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution.
Faculty rank (professor, associate professor, assistant professor, and lecturer) is an example of discrete numerical data.
The responses to the question “How many movies have you seen in the last month?” are values from a discrete variable.
For each trial in a binomial random variable the probability of success is constant.
A binomial distribution with ? =0.2 is always right skewed.
A Poisson distribution will be left-skewed for small values of its parameter, ?.
In a Poisson distribution, the mean and variance are equal.
If ? remains constant in a Binomial distribution, an increase in n will not change the mean.
Multiple Choice Questions
Which of the following is a characteristic of every binomial distribution?
A. Each outcome is dependent on the previous outcome.
B. The probability of success increases from trial to trial.
C. Each outcome is mutually exclusive.
D. The outcome of a trial depends on the number of trials.
Ten bolts were selected randomly from a production line and the diameter of each was measured. Why do these diameters not have a binomial distribution?
A. They have equal probability of success.
B. The diameters of the bolts are not independent of one another.
C. Diameter is a continuous random variable.
D. They do have a binomial distribution.
Which of the following cannot generate a Poisson distribution?
A. The number of people at a movie theatre on Friday evening
B. The number of customers arriving at an ATM in a minute.
C. The number of defects in a new automobile.
D. The number of goals in the World Cup soccer game.
The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 5.5 tickets per day. Interpret the value of the mean.
A. The number of tickets that is written most often is 5.5 tickets per day
B. Half of the days have less than 5.5 tickets written and half of the days have more than 5.5 tickets written
C. If we sampled all days, the arithmetic average or expected number of tickets written would 5.5 tickets per day.
D. The mean cannot be interpreted.
A multiple-choice test has 20 questions, and there are 4 choices for each question. A student comes to class with four different colored balls of identical shape and size in her pocket. She has decided to draw one ball from her pocket per question and allow the color of the ball to select the answer for each question. What type of probability distribution can be used to figure out her chance of getting 10 questions right?
A. Hypergeometric distribution
B. Poisson distribution
C. Normal distribution
D. Binomial distribution
The Information Commons in the main library has 150 personal computers. The probability that any one of them will require repair on a given day is 0.02. To find the probability that exactly 25 of the computers will require repair, one will use what type of probability distribution?
A. Binomial distribution
B. Poisson distribution
C. Normal distribution
D. Hypergeometric distribution
Buyers for large department stores typically purchase costume jewelry in assortments rather than by the piece. A buyer is interested in ordering a ring that is made in 5 sizes, and comes in lots of 50 rings. The sizes and the proportion of rings manufactured in that size are in the table below. Use this information to answer the next THREE questions.
Size
5
6
7
8
9
Probability
0.1
0.2
0.4
0.2
0.1
What is the probability that, at most, a shipment received will contain 10 size six rings?
A. =BINOMDIST(10,50,0.2,1)
B. =1-BINOMDIST(10,50,0.2,1)
C. =BINOMDIST(11,50,0.2,0)
D. =1-BINOMDIST(10,50,0.2,0)
What is the probability that a shipment of 50 rings will have more than 20 but no more than 30 size 7 or size 8 rings?
A. =BINOMDIST(10,50,0.6,1)
B. =BINOMDIST(20,50,0.6,1)+BINOMDIST(30,50,0.6,1)
C. =BINOMDIST(30,50,0.6,1)-BINOMDIST(19,50,0.6,1)
D. =BINOMDIST(30,50,0.6,1)-BINOMDIST(20,50,0.6,1)
What is the expected number of size 9 rings in a 50 ring shipment?
A. 7 B. 9 C. 5 D. 3.5
Whenever ? = 0.5, the binomial distribution will:
A. be symmetric only if n is large.
B. always be symmetric.
C. be right-skewed.
D. be left-skewed.
Two international students were talking in the wake of the Singapore Air disaster a few years ago. The first student said she had been intending to fly Singapore Air for the semester break, but wasn’t going to now that there had been an accident! The second said, “But it should be extra safe now. The chances of it crashing again so soon are very small.” If the likelihood of an airplane crashing follows a Poisson distribution, what are these students forgetting about a Poisson random variable?
the Poisson random variable is discrete.
the number of trials must be at least 30.
the outcome of each trial is one of two mutually exclusive categories.
the results of one trial is not dependent on the results of the other trials.
Which of these parameters associated with a binomial experiment will produce a probability distribution with the smallest standard deviation, given n is the same for each situation?
A. ? = 0.5 B. ? = 0.4 C. ? = 0.3 D. (1- ?) = 0.1.
Past history shows that 70% of college students prefer Pepsi. Groups of nine students are selected. Describe the histogram that would represent this distribution. The histogram would be:
A. Left-skewed with ten columns.
B. Right-skewed with nine columns.
C. Left-skewed with nine columns.
D. Right-skewed with ten columns.
Solved Problems from Text
6.1 A is a probability distribution, since the sum of P(x) is 1 and all probabilities are nonnegative, while B and C are not probability distributions since the sum of P(x) is .95 for B and 1.30 for C.
6.2 a. E(X) = 70, V(X) = 100, ? = 10. The worksheet is:
x
P(x)
xP(x)
xE(X)
(x-E(X))2
P(x)[xE(X)]2
60
0.40
24.00
-10.00
100
40.00
70
0.30
21.00
0.00
0
0.00
80
0.20
16.00
10.00
100
20.00
90
0.10
9.00
20.00
400
40.00
Total
1.00
?=E(X)=70.00
?=V(X)=100.00
b. Distribution is skewed to the right. Here is the graph of the distribution, with the long right tail.
Apply the formulas for mean and standard deviation of a Uniform distribution.
Using a = 0000 and b = 9999,
Using a = 0000 and b = 9999,
First, define success and assign it a value of 1. Now define failure as everything that is NOT success and assign it a value of 0. Remember, in order to be a valid Bernoulli trial, there can only be two mutually exclusive and collectively exhaustive outcomes. Your answers may vary which would switch 0 and 1.
a. 1 = correct, 0 = incorrect
b. 1= insured, 0 = uninsured
c. 1 = busy, 0 = not busy
d. 1 = lost weight, 0 no weight loss
6.15 Apply the formulas for the mean and standard deviation of a binomial distribution.
and
a. ? = (8)(.1) = 0.8,
b. ? = (10)(.4) = 4,
c. ? = (12)(.5) = 6,
d. ? = (30)(.9) = 27,
e. ? = (80)(.7) = 56,
f. ? = (20)(.8) = 16,
6.16 Using the Excel BINOMDIST function =BINOMDIST(k,n,?,0), where k = the value of X you want the probability for, n =number of trials, ? = probability of success and 0 means you want only the probability for the one X value,
a. P(X = 2) =BINOMDIST(2, 8, .10, 0) = .1488
b. P(X = 1) =BINOMDIST(1, 10, .40, 0) = .0403
c. P(X = 3) =BINOMDIST(3, 10, .40,0) = .0015
d. P(X = 5) =BINOMDIST(5, 9, .90, 0) = 0.0074
6.23 Remember, the mean and variance of a Poisson are identical and equal to lambda, ?. Thus, in each case above, the mean = ? and the standard deviation = .
a. ? = 1, ? = 1.0 and ? =1
b. ? = 2, ? = 2.0 and ? =1.414
c. ? = 4, ? = 4.0 and ? =2.0
d. ? = 9, ? = 9.0 and ? =3
e. ? = 12, ? = 12.0 and ? =3.464
6.24 The Excel formula for calculating Poisson probabilities is =POISSON(x,?,0), where, as in the binomial, x is the value of the random variable in which you are interested, ? is the value of the paramater, and 0 indicates that you only what the probability associated with a specific value of the random variable.
a. ? = 0.1, P(X = 2) =POISSON(2, 0.1, 0) = .24377
b. ? = 2.2, P(X = 1) =POISSON(1, 2.2, 0) =.00452
c. ? = 1.6, P(X = 3) = POISSON(3, 1.6, 0) = .13783
d. ? = 4.0, P(X = 6) = POISSON(6, 4.0, 0) = .10420
e. ? = 12.0, P(X = 10) =POISSON(10, 12, 0) = .10484
Quiz Yourself Answers
True/False
Multiple Choice
1
T
6
F
1
C
6
B
11
D
2
F
7
T
2
C
7
A
12
D
3
T
8
F
3
A
8
D
13
A
4
T
4
C
9
C
5
F
5
D
10
B
