The Minimum Ball Number

Java Applet Simulation of the ball and urn experiment

In the ball and urn experiment, let U denote the random variable that gives the smallest number of a ball in the sample:

V = max{X₁, X₂, ..., X_n}

where X_i is the number of the i'th ball selected. In the exercises below, use techniques similar to the ones used for the maximum ball number V:

Sampling With Replacement

$Mathematical Exercise$ 1. If the sampling is with replacement, show that the density function of U is given by

2. In the ball and urn experiment, select sampling with replacement and random variable U. Vary the parameters and note the shape of the graph of the density function. Now let N = 50 and n = 10 and run the experiment with an update frequency of 100. Note the apparent convergence of the relative frequency function to the density function.

$Mathematical Exercise$ 3. For fixed n and k, show that

Interpret the result.

$Mathematical Exercise$ 4. For fixed N, show that as n increases to infinity,

Interpret the result.

Sampling Without Replacement

$Mathematical Exercise$ 5. If the sampling is without replacement, show that the density function of U is given by

6. In the urn experiment, select sampling without replacement and random variable U. Vary the parameters and note the shape of the graph of the density function. Now let N = 50 and n = 20 and run the experiment with an update frequency of 100. Note the apparent convergence of the relative frequency function to the density function.

An Estimator of N Based on U

$Mathematical Exercise$ 7. If the sampling is without replacement, show that the expected value of U is

E(U) = (N + 1) / (n + 1)

It follows from Exercise 7 that

(n + 1)U - 1

is an unbiased estimator of N.

8. In the urn experiment, choose sampling without replacement and set N = 100, R = 40, and n = 20. Run the experiment 50 times, updating after each run. For each run, compute the estimate of N based on U, the estimate of N based on V, and the estimate of N based on Y. For each estimator, compute the square root of the average of the squares of the errors over the 50 runs. Based on these empirical error estimates, rank the estimators of N in terms of quality.