Assume that we have three independent observations: where Xi ~ Binomial(n = 7,p) for i {1, 2, 3). The value of p (0, 1) is not known When we have observations like this from different, independent random variables, we can find joint probabilities by multiplying together the individual probabilities.

9 / 10^ Figure 1 : Plot for question S, part ( c). You may assume that the likelihood attains a maximum in the { ( I mark ; Find the maximum likelihood estimate of' p by using the data: $1 = 2, `` = O. K. = 3. Note that in = 3. Compare the result with the one that you have obtained in part ( c) . h ( & marks ; Using the expression for ; obtained in part ('f') , find Help^ and Valley , fully justifying all your working . Is jan unhinged! estimator of } } Explain your answer .Assume that we have three independent observations: X1=21 X2=51 X3=31 where X, ~ Binomial(n = 7,33) for i E {1,2,3}. The value ofp 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 the individual probabilities. For example, P(X1= $1,152 = I21X3 = $3) = P(X1= I1)P(X2 = 272)]?(X3 = '33)- 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 X1, 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 Q_ 9 _ 10 _ dpKp (1 p) (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 0. f (5 marks) Find %, and give all possible solutions to the equation dL 5 = 0. Show that the maximum likelihood estimatg for p is X1+X2+\"'+Xm p mn