Question

Assume that we have three independent observations: where Xi ~ Binomial(n 7,p) for i E { 1.2.3). The value of p E (0, 1) is not known. When we have observations like this from different, independent ran- dom variables, we can find joint probabilities by multiplying together th ndividual probabilities. For example This should remind you the discussion on statistical independence of random variables that can be found in the course book (see page 22) Answer the following questions a (4 marks) Consider that Xi, X2, X3 are independent Binomial(7,p) random variables, as above Write down the likelihood function, L(p: 2,5, 3). State the range of values of p for which the likelihood function is defined b (4 marks) Show that dL dp = Kp9(1-p)10(10-21p), where K is a constant that you should specify (but not calculate) c (3 marks) The graph of the likelihood function for 0< P<1 is given in Figure 1. Find the maximum likelihood estimate of p. Remember to make reference to the graph in justifying your answer d (1 mark) Write a sentence in English to explain what the maxi mum likelihood estimate from (c) represents (see page 57 in the course book) e (4 marks) Let m be an integer with the property that m 2 2. Consider that Xi,X2,... , Xm are independent Binomial(n,p) random variables, where n is known and p is unknown. Note that p (0,1). Write down the expression of the likelihood function dL f (5 marks) Find dn and give all possible solutions to the equation dL 0. Show that the maximum likelihood estimator for p is mn

Answer 1

a. 0 0 2-1 Scanned with

0 cl Ca 引 0 Scanned with Camscanne

B A 7 2 rm 九 Scanned with Camscanne