**Final**

**Math 286/366**

**Spring, 2001**

Relevant entries from the standard normal table (90%,95%,97.5%) = (1.28, 1.645, 1.96).

For **ALL**
simulations required by this exam, you are to use the following random samples,
in order. The symbols U_{1}^{
} and U_{2} denote random
numbers.

-ln(U |
.64 |
.28 |
.42 |
.30 |
1.53 |
.96 |

U |
.50 |
.19 |
.26 |
.61 |
.87 |
.22 |

1. You want to simulate, in the most efficient way, arrivals from 1:00 pm to 2:00 pm from a non-homogeneous Poisson process with l(t) = 2t, 0 ² t ² 1. Under the completed simulation (using thinning), when does the second arrival occur?

2. After 39 simulations of a distribution, you obtain a sample mean and sample variance

If the 40^{th} simulation produces a value of 33,
what is the length of a 90% confidence interval for the mean, based on your 40
simulations?

3. You obtain the following sample from an unknown distribution: 20,21,24,25,26,28. To generate two bootstrap samples you roll two sequences with a die: Sequence 1: 1,1,3,3,4,6; Sequence II: 1,2,2,3,5,5. Use these to estimate the mean square error involved in estimating the mean of the unknown distribution with the sample mean.

4. You wish to test that two samples are from the same distribution using the rank sum test. Sample I: 1.5, 2.7, 5.7; Sample II: 0.3, 1.7, 4.7. What is the (exact) p-value you obtain?

5. You
wish to test whether or not a sample 0.48, 0.76, 0.87, 0.96 comes from the
distribution with probability density function f(x) = 2x, 0 ² x ² 1.
What value of

D does the Kolmogorov-Smirnoff test produce?

6. An insurer experiences claims that come in according to a Poisson process with an expectation of 3 claims per year. Claims are paid immediately. The amount of a claim has a uniform distribution on the interval (0,2) and the interest rate is 10%. Produce one simulation of the present value of claims during the next year.