# Homework 2 ISyE 6420 2-D Density Tasks solution

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Original Work ?

## Description

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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).

A lifetime X (in years) of a particular device is modeled by a
Weibull distribution
f(x|ν, θ) = νθxν−1
exp {−θxν
} , x ≥ 0,
with shape parameter ν = 3 and unknown rate parameter θ. The lifetimes of X1 = 2, X2 = 3,
and X3 = 2 are observed. Assume that an expert familiar with this type of devices suggested
an exponential prior on θ with rate parameter 2.

(a) For the prior suggested by the expert, find the posterior distribution of θ.
(b) What are the posterior mean and variance? No need to integrate if you recognize to
which family of distributions the posterior belongs.

## 3. 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(λ) distribution, where λ is the rate parameter.
(a) Suppose λ = 1/5, find the probabilities that
(i) a run continues for at least 5 hours.
(ii) a run lasts less than 10 hours.
(iii) a run continues for at least 10 hours, given that it has lasted 5 hours.

(b) Now suppose that the rate parameter λ is unknown, but there are three measurements
of interblockage times, T1 = 2, T2 = 4, and T3 = 8.
(i) How would classical statistician estimate λ ?
(ii) What is the Bayes estimator of λ if the prior is π(λ) = √
1
λ
, λ > 0.

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