Remember to state the range of values of β for which the log- likelihood is defined. Show your working.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

4 Suppose that X1, X2,. .., Xn are n = 10 independent Bernoulli ran- dom variables with the property that 

P(Xi    1

= 1) = , for i 1, 2,. .., n ,

β + 1  

where 0 < β < 1. The value of β is unknown. For estimating the value of β, we organize an experiment in which we obtain Xi = xi for i 2 {1, 2,. .., n}, as follows:

x1 = x2 = 1, x7 = 1,

x3 = x4 = x5 = x6 = 0, x8 = x9 = x10 = 0.

Note that n1 = 3 of the values x1, x2,. .., xn are ones and the other

n — n1 = 7 are zeros.

Answer the following questions:

a (5 marks) Write down the log-likelihood function for this problem, log L(β; x1, x2,. .., xn).

Remember to state the range of values of β for which the log- likelihood is defined. Show your working.

b (4 marks) Find the maximum likelihood estimate of β. In your answer, you should di↵erentiate the log-likelihood with respect to the parameter β, and set to 0 for the maximum. Quote your numerical answer to 4 decimal places.

p = 1/β + 1 

The fact 0 < β < 1 leads to 0 <p < 1.

c (4 marks) If we employ the definition above, then the unknown parameter is p. Assuming that the measurements are the same x1, x2,. .., xn, show that the log-likelihood function has the ex- pression:

log L(p; x1, x2,. .., xn) = n1 log(p) + (n — n1) log(1 — p).

Remember to state the range of values of p for which the log- likelihood is defined. Show your working.

d (4 marks) Find the maximum likelihood estimate of p. In your answer, you should di↵erentiate the log-likelihood with respect to the parameter p, and set to 0 for the maximum. Quote your numerical answer to 1 decimal place.

You should make reference to the graph of the log-likelihood func- tion, which is shown in Figure 5. Show all your working in your answer.

e (1 mark) Use the result obtained in part (d) in order to write down the expression of the maximum likelihood estimator p.

f (1 mark) Use the value of p computed in part (d) for finding β (see  again  the  definition  of  p).  Compare  the  value  of  βb with  the on obtained in part (b).

Total: 55 marks.

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