# Homework 3 ISyE 6420 Estimating the Precision solution

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## Estimating the Precision Parameter of a Rayleigh Distribution.

If two random
variables X and Y are independent of each other and normally distributed with variances
equal to σ
2
, then the variable R =

X2 + Y
2
follows the Rayleigh distribution.

Parameterized with precision parameter ξ =
1
σ2 , the Rayleigh random variable R has a density
f(r) = ξr exp (

ξr2
2
)
, r ≥ 0, ξ > 0.

An example of such random variable would be the distance of darts from the target center
in a dart-throwing game where the deviations in the two dimensions of the target plane are
independent and normally distributed.

(a) Assume that the prior on ξ is exponential with the rate parameter λ. Show that the
posterior is gamma Ga

2, λ +
r
2
2

.

(b) Assume that R1 = 3, R2 = 4, R3 = 2, and R4 = 5 are Rayleigh-distributed random
observations representing the distance of a dart from the center. Find the posterior in this
case for the same prior form (a), and give a Bayesian estimate of ξ.

(c) For λ = 1, numerically find 95% Credible Set for ξ .
Hint: In (b) show that if r1, r2, . . . , rn are observed, and the prior on ξ is exponential E(λ),
then the posterior is gamma Ga

n + 1, λ +
1
2
Pn
i=1 r
2
i

.

## 2. Estimating Chemotherapy Response Rates.

An oncologist believes that 90% of
cancer patients will respond to a new chemotherapy treatment and that it is unlikely that this
proportion will be below 80%. Elicit a beta prior on proportion that models the oncologist’s
beliefs.
Hint: For elicitation of the prior use µ = 0.9, µ − 2σ = 0.8 and expressions for µ and σ
for beta.
During a trial, in 30 patients treated, 22 responded.

(a) What are the likelihood and posterior distributions? What is the Bayes estimator of
the proportion?
(b) Using Octave, R, or Python, find 95% Credible Set for p.

(c) Using Octave, R, or Python, test the hypothesis H0 : p ≥ 4/5 against the alternative
H1 : p < 4/5.
(d) Using WinBUGS, find the Bayes estimator and Credible Set and conduct the test.
Compare WinBUGS results with (a-c).
2