MBS

4.57 (30 points)

c. For a poisson random variable, we have:

So, to calculate the given probability we would have to evaluate the following sum:

Which, incidentally, gives P(x£122)»0.3

d. The poisson model requires that the probability of the event occuring in a given unit of time is the same for all units. Thus, if the probability of a bank failure differed from one year to the next between 1988 and 1994, the poisson model would be inappropriate. Also, the number of bank failures in each year must be independent of the number in other years.

7.75 (20 points)

Because the sample size n is large relative to the population size N, it is necessary to adjust the standard error of the estimator by a finite population correction factor:

Thus a 95% confidence interval is given by:

Minitab Macro (25 points)

Here is the macro file:

GMACRO

Poisson

# k1=counter

# k2=number of iterations

# k3=sample size

# k4=lambda

Let k5=1/k4

Do k1=1:k3

Let C3(k1)=0

Enddo

name C2 'Time'

name C3 'Event'

Do k1=1:k2

Random k3 C1;

Exponential k5.

Let C2 = PARS(C1)

Plot C3*C2;

Symbol;

Title "Poisson Process";

ScFrame;

ScAnnotation.

Enddo

ENDMACRO

It is executed with the following commands in the command line editor:

let k2=5

let k3=10

let k4=.83

%d:\90-786\poisson

Here are the plots:

Chattergee – Mortgage Rates (25 points)

Here are the descriptive statistics by type:

Descriptive Statistics

Variable 0=Fixed N Mean Median TrMean StDev

Rate 0 14 7.357 7.313 7.354 0.404

1 6 4.917 4.750 4.917 0.645

Variable 0=Fixed SE Mean Minimum Maximum Q1 Q3

Rate 0 0.108 6.750 8.000 7.062 7.594

1 0.264 4.250 6.000 4.438 5.437

The confidence interval is constructed as:

Where t_a/2 is based on (n-1) degrees of freedom.

For fixed rate mortgages we have:

For variable rate mortgages: