# Homework 2 ISyE 6420 Cell Clusters in 3D Petri Dishes solution

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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.

If
f(x, y) = (
λ
2
e
−λy
, 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) = √
1
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.
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