Download 8.3 olling a fair die: the discrete uniform distribution

3. Using Table 8.3, determine the following theoretical Poisson probabilities:
a. The probability of 0 occurrences when the
mean is 0.7
b. The probability of 3 occurrences when the
mean is 5
c. The probability of 11 occurrences when the
mean is 5
d. The probability of 1 occurrence when the
mean is 0.1
4. At a fast-food restaurant, the arrival of
customers during the period from 11:00
A.M. to 2:00 P.M. is tracked. The owner
is interested in finding out whether or
not the arrival of customers follows the
Poisson distribution. Go through the three
conditions for a Poisson random variable.
Which of those three is most clearly being violated in this case? Explain.
5. State the relationship between the theoretical
standard deviation and the theoretical mean
of a Poisson random variable. What is the
theoretical standard deviation in the case of
the Prussian army data?
6. The following table shows probabilities obtained in an experiment. The table also has the
Number of
occurrences
Experimental
probability
Theoretical
probability
0
1
2
3
4
5
6
7
0.053
0.153
0.138
0.103
0.225
0.214
0.068
0.047
0.033
0.113
0.193
0.219
0.186
0.126
0.072
0.035
corresponding probabilities under the Poisson
distribution with a mean of 3.4.
a. Make a histogram of the experimental
probabilities, and a histogram of the theoretical probabilities.
b. Compare the two histograms. Do the obtained data seem to follow the Poisson
distribution with mean 3.4?
7. Reconsider the data given in Exercise 6. Recall
that in Chapter 7 we discussed the chi-square
test, a test used to determined whether data
follow a certain distribution. Assume that in
the experiment described Exercise 6, the experimental probabilities were actually based
on 1000 observations.
a. Reconstruct the results of the experiment,
giving the number of outcomes for each of
the values. For example, there were 1000 ⫻
0.053 ⳱ 53 outcomes of 0.
b. Using the theoretical probabilities, calculate how many of the 1000 observations
are expected for each of the values. For example, we would expect 1000 ⫻ 0.033 ⳱ 33
outcomes of 0.
c. Calculate the value of the chi-square statistic for these data.
d. What are your conclusions based on the
chi-square test?
8. Using Table 8.3, determine the following theoretical Poisson probabilities:
a. The probability of either 1, 2, or 3 occurrences when the mean is 0.1.
b. The probability of at least 2 occurrences
when the mean is 5.
c. The probability of at most 3 occurrences
when the mean is 1.
8.3 ROLLING A FAIR DIE: THE DISCRETE
UNIFORM DISTRIBUTION
If you roll a fair six-sided die, as has been done often in this book, you are
equally likely to get 1, 2, 3, 4, 5, or 6. There is a name for such a distribution:
the discrete uniform distribution from 1 to 6. Discrete means that the
possible values are specific values that can be listed. (The opposite of discrete
is continuous, which we discuss in Section 8.5 for a uniform distribution.)
Uniform means that each of the values is equally likely. Since there are six
values in the case of a normal die, the theoretical probability of any one of
them appearing is 16 . A 32-sided die, if it is fair, will yield a discrete uniform
distribution with outcomes ranging from 1 to 32. It is not necessary to start
with 1. For example, a five-sided die could have the numbers ⫺2, ⫺1, 0, 1,
2. Its distribution will be a discrete uniform distribution ranging from ⫺2
to 2. One needs only to specify the lowest and the highest integer. Also, use
of a “die” here is unnecessary. A die is merely a tangible physical object
to help us imagine generating the data. The essence of the discrete uniform
distribution is a sequence of equally spaced integers each having the same
theoretical probability.
The theoretical mean is just halfway between the low value and the
high value:
low value Ⳮ high value
2
For example, the theoretical mean for a fair six-sided die is (low valueⳭhigh
value)/2 ⳱ (1 Ⳮ 6)/2 ⳱ 3.5.
The formula for the theoretical standard deviation is not very intuitive.
For completeness, we give it:
Theoretical mean ⳱
Theoretical standard deviation ⳱
冪
(high value ⫺ low value Ⳮ 1)2 ⫺ 1
12
For the six-sided die, this standard deviation is
冪 (6 ⫺ 1 Ⳮ121) ⫺ 1 ⳱ 冪 6 12⫺ 1
36 ⫺ 1
⳱冪
⳱ 1.7078
12
Standard deviation ⳱
2
2
Thus there will be quite a bit of variability from the average of 3.5 in that
the typical variation is roughly from 2 to 5.
SECTION 8.3 EXERCISES
1. For each of the following discrete uniform
distributions, give the value of the theoretical
mean and the theoretical standard deviation.
a. High value ⳱ 10, low value ⳱ 1
b. High value ⳱ 8, low value ⳱ 4
c. High value ⳱ 5, low value ⳱ ⫺5
d. High value ⳱ 15, low value ⳱ 6
2. Repeat Exercise 1 for these values:
a. High value ⳱ ⫺5, low value ⳱ ⫺9
b. High value ⳱ 21, low value ⳱ 10
c. High value ⳱ 0, low value ⳱ ⫺10
d. High value ⳱ 32, low value ⳱ 25
3. You and three of your friends agree to meet
for lunch every day. You begin to keep track
of how many of your friends beat you to the
meeting place each day. You suspect that this
random variable has a uniform distribution.
a. What are the possible values of the random
variable?
b. Assuming that this random variable does
follow the uniform distribution, what are
its theoretical mean and standard deviation?
c. The actual data are as follows: Out of a
total of 50 meetings . . .
There were 10 times when none of your
friends were there when you arrived.
There were 13 times when one of your
friends was there when you arrived.
There were 15 times when two of your
friends were there when you arrived.
There were 12 times when all three of your
friends were there when you arrived.
Use the chi-square test as described in
Chapter 7 to determine whether these observed data do indeed follow the uniform
distribution.
8.4 CONTINUOUS DISTRIBUTIONS
The binomial and Poisson distributions are both based on counting; the
discrete uniform distribution deals with a list of consecutive integers.
These are all discrete distributions. It is common to find data based on
measurements instead of counts: for example, mountain heights, light bulb
lifetimes, incomes, agricultural yields, and so on. In many measurements,
we may be able to specify the smallest and largest possible values, but any
value in between those, not just integers, may be possible. For example,
a lightbulb’s lifetime (the length of time it is continuously on before
breaking) cannot be less than 0 hours, although if it is defective it may
last exactly 0 hours. There is some maximum length of time—if not 1000
hours, then maybe 10,000 hours or more. But any value between 0 and
some large maximum is possible. It may last exactly one hour, or half
an hour, but it is more likely to last a less rounded-off time, such as
134.398459987 hours. Of course, in practice one cannot find the exact
lifetime, but instead finds it to the nearest minute, second, or maybe 10th
of a second. In any case, the actual number of possible values is immense;
theoretically, it is infinite if one ignores the fact that all measurements are
1
rounded off (for example, a length might be rounded to the nearest 1000
of
an inch).
Moreover, measurements are often best viewed as random, in which
case it is necessary to develop probability distributions to describe them.
For example, we might be randomly sampling measurements of a real
population, such as heights of adult females, in which case height can be
assigned a continuous probability distribution. Or we might be randomly
sampling from a potentially infinitely repeatable experimental process, such
as assessing the lifetime of lightbulbs as they come off the production line,