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STAT 3001/7301 Mathematical Statistics

Tutorial Sheet 8

2023

1. Find a conjugate prior for the Exp(θ) distribution.

2. Let x1 ,x2 , . . . ,xn  be an iid sample from Exp(1/θ). Find a conjugate prior for this distribution and determine the posterior distribution.

3. NOTE: This question requires the use of computer software.

There are a few options for ordering food at a local restaurant.  Yesterday, the number of orders placed at this restaurant is as follows:

Note that take-away orders are placed in-store and pick-up orders are placed either on the phone or online.

(a) Suppose we assume the data comes from a multinomial distribution and we

use a Dirichlet prior p  ∼  Dirichlet(α) on the underlying proportion p  = (p1 ,p2 ,p3 ,p4 ) for the four ordering methods, where α = (α1 ,α2 ,α3 ,α4 ). Give an expression for the posterior distribution of p given the above data and identify the distribution.

(b) The data from last week’s sales were

Describe how you might use this information to choose a value for the hyper- parameter α .

(c) By simulating from your posterior distribution, or otherwise, provide point estimates and associated 95% (posterior) credible intervals for

i. the proportion p1  of customers who dine-in

ii. the proportion p4  of customers who use delivery

iii. the proportion of customers who come in-store

iv. the difference in the proportions of dine-in and delivery orders