Question

Consider a large population of individuals and let θ denote the (unknown) proportion of the population belonging to a sensitive group A (e.g. drug users). Suppose, we randomly select n individuals from the population and ask each person to select a card from a deck and answer the question written on the card. Each card in
the deck has one of the two questions: Q1: Do you belong to A? and Q2: Do you not belong to A? Also, 85% percent of the cards ask Q1 and the remaining 15% ask Q2. Assume that each person answers Yes or No truthfully to the selected question. For i = 1,...,n, let Xi = 1 if the ith person answers ‘Yes’ otherwise Xi = 0. So, the data are the observed values of X1,...,Xn. Give the joint distribution of X1,...,Xn and the distribution of the total number of Yes responses.

Answer 1