Answers to Problems in Section 9 of the Supplement

Answers to Problems in Section 9 of the Supplement

In these answers, when we take the logarithm we always use the natural logarithm, Hence, log(x) means the natural logarithm of x.
PROBLEM 1. .

We need to maximize with respect to . It’s easier to maximize the logarithm of instead. The value of that maximizes the will also maximize . So, let’s take the derivative of .
. We set this equal to zero and solve for to get:

PROBLEM 2. These data are from the same probability density function that we used in problem 1, just making the x into a y and the into an. Hence the MLE of equals

PROBLEM 3. We need to maximize with respect to . It’s easier to maximize the logarithm of instead. The value of that maximizes the will also maximize . So, let’s take the derivative of .
.

We set this equal to zero and solve for to get:

Using the given data, the MLE = 4/(3+5+6+10) = 1/6.
PROBLEM 4

For 25 people, plug in 25 for n.
5. Let Y = the number of occupants in a car. Then,

The last probability is obtained from formula for the sum of an infinite geometric series.
To obtain the likelihood function, we have to multiply the probabilities for all 1011 cars that were observed. Since there are 678 cars with Y=0, 227 cars with Y-1, etc., the likelihood function for p is:

We then need to take the derivative of this likelihood function (let’s use the logarithm of the likelihood function instead), and set it equal to zero. We obtain:
.
Solving for p, we get (997+525)p=997, so that the MLE for p is 997/(997+525) = .655.
PROBLEM 6.

We need to maximize with respect to . It’s easier to maximize the logarithm of instead. The value of that maximizes the will also maximize . So, let’s take the derivative of
.
. We set this equal to zero and solve for to get: