You should make reference to the graph of the likelihood function, which is shown in Figure 2. Show all your working in your answer

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

2 Suppose that each of two. tatisticians A and B must estimate a certain parmneter p whose value is miknown (0 < p < 1). 

Statistician A can observe the value of a random variable X, which has the Negative Binomial distribution with parameters k = 4 and p. The value observed by statistician A is X = 6. 

(a). (6 marks) Write down the likelihood function for this problem, Lp(p;13). Remember to state the range of values of p for which the likelihood la defined. 

Find the maximum likelihood estimate of p. In your answer, you should differentiate the 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 likelihood function, which is shown in Figure 2. Show all your working in your answer. 

Statistician B can observe the value of a random variable Y, which has the Binomial distribution with parameters n = 10 and p. Suppose that the value observed by statistician B la Y = 4. 

(b). (1 mark) W.te down the log-likelihood function for this problem, log En (p; 4). Remember to state the range of values of p for which the log-likelihood A defined. In your answer, you C211 use the expression of the likelihood function given in Section 3.3.2 of the comae book, without proving it. (No. lug =1uy = 

(c). (3 marks) Use the results obtained in part (a) and in part (b) to show that there exits a constant C (which does not depend w p) such that log Lu(p: 4) = log Lk(p:6)+C, for all p e (0,1). Write down the expression of C and compute its numerical value. Quote your answer to 2 decimal places. d (5 marks) Use the results obtained in pan (a) and in part by bh order to find the value off. with the paucity that 

6.04 4) 0 bn(pi 4). 11,2 all p 6 (0,1).

Explain your answer. e (2 marks) For statistician B, compute fiby applying the fccmula for the marirrann likelihood esthnae giver in Section 3.3.2 of the course book. Decide if the .alma of p ralektlated in this way is equal to the value of (3 obtained iu part

(d). 3 Suppose that a scientist desires to estimate the proportion p of monarch butterflies that have a special type of marking on their wings. 

Suppose that he captures monarch butterflies one at a time until he has found one that has this special marking. 

Let X be the number of butterflies raptured by the scientist before he has captured the butterfly with the special mark. 

Answer the following questions: 

a) (2 marks) State the distribution of X, with parameters. In your answer, assume that the captures are independent and the probability of success is p for each of them. Note that 0 < p < 1. b (7 marks) Assume that the scientist must capture a total of 43 butterflies. Write down the likelihood function for this problem, L(p; x), being careful with the observed value of X. Remember to state the range of values of p for which the likelihood is defined. Find the maximum likelihood estimate of p. Quote your numer-ical answer to 4 decimal places. You should make reference to the graph of the likelihood function, which is shown in Figure 3. Show all your working hi your answer. 9 xfo. 

0 < p < 1.

b) (7 marks) Assume that the scientist must capture a totalof 43 butterflies. Write  down the likelihood function for this problem, L(p;x), being careful with the observed value of X. Remember to state the range of values of p for which the likelihoodis defined. Find the maximum likelihood estimate of  p. Quote  your numerical answer o 4 decimal places. You should mmake reference to the graph oof the lilke function.

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