Homework 2 ISyE 6420 Cell Clusters in 3D


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1. Cell Clusters in 3D Petri Dishes.

The number of cell clusters in a 3D Petri dish
has a Poisson distribution with mean λ = 5. Find the percentage of Petri dishes that have
(a) 0 clusters, (b) at least one cluster, (c) more than 8 clusters, and (d) between 4 and 6
clusters inclusive.

2. Silver-Coated Nylon Fiber.

Silver-coated nylon fiber is used in hospitals for its
anti-static electricity properties, as well as for antibacterial and antimycotic effects. In the
production of silver-coated nylon fibers, the extrusion process is interrupted from time to
time by blockages occurring in the extrusion dyes. The time in hours between blockages, T,
has an exponential E(1/10) distribution, where 1/10 is the rate parameter.

Find the probabilities that
(a) a run continues for at least 10 hours,
(b) a run lasts less than 15 hours, and
(c) a run continues for at least 20 hours, given that it has lasted 10 hours.
If you use software, be careful about the parametrization of exponentials.

3. 2-D Density Tasks.

f(x, y) = (
, 0 ≤ x ≤ y, λ > 0
0, else

Show that:
(a) marginal distribution fX(x) is exponential E(λ).
(b) marginal distribution fY (y) is Gamma Ga(2, λ).
(c) conditional distribution f(y|x) is shifted exponential, f(y|x) = λe−λ(y−x)
, y ≥ x.
(d) conditional distribution f(x|y) is uniform U(0, y).

4. Nylon Fiber Continued.

In the Exercise 2, the times (in hours) between blockages
of the extrusion process, T, had an exponential E(λ) distribution. Suppose that the rate
parameter λ is unknown, but there are three measurements of interblockage times, T1 = 3,
T2 = 13, and T3 = 8.

(a) How would classical statistician estimate λ?
(b) What is the Bayes estimator of λ if the prior is π(λ) = √
, λ > 0.

Hint: In (b) the prior is not a proper distribution, but the posterior is. Identify the posterior
from the product of the likelihood from (a) and the prior, no need to integrate.