Feller (1957) includes a number of examples of observations fitting the Poisson distribution, including data on the number of flying-bomb hits in the south of London during World War II. The city was divided into 576 small areas of one-quarter square kilometers each, and the number of areas hit exactly k times was counted. The observed frequencies are given in the table below.

(a)  Find the likelihood function for this observed data (you may ignore the terms that are not important for maximum likelihood estimation).

(b)  Write R code to plot the likelihood function (for λ ∈ (0.6, 1.2))

(c)  Find the maximum likelihood estimate of λ (Please do NOT use calculus!)

(d)  Use the likelihood ratio test to test the null hypothesis H0 : λ = 1 against the alternative H1 : λ != 1

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Q. 6 (20 pts) (5 pts each) Feller (1957) includes a number of examples of ob- servations fitting the Poisson distribution, including data on the number of flying-bomb hits in the south of London during World War II. The city was divided into 576 small areas of one-quarter square kilometers each, and the number of areas hit exactly k times was counted. The observed frequencies are given in the table below. Hits 0 1 Frequency 229 211 2 3 4 5 211 93 35 7 1 Assume that the number of hits in a small area (of one-quarter square kilo- meters) has a Poisson distribution with parameter . Note that the Poisson p.m.f, P(Y = y) = = e-11 y! (a) Find the likelihood function for this observed data (you may ignore the terms that are not important for maximum likelihood estimation). (b) Write an R code to plot the likelihood function (for (0.6, 1.2)) (c) Find the maximum likelihood estimate of X (Please do NOT use calcu- lus!) (d) Use the likelihood ratio test to test the null hypothesis Ho : X = 1 against the alternative H : 1 Q. 6 (20 pts) (5 pts each) Feller (1957) includes a number of examples of ob- servations fitting the Poisson distribution, including data on the number of flying-bomb hits in the south of London during World War II. The city was divided into 576 small areas of one-quarter square kilometers each, and the number of areas hit exactly k times was counted. The observed frequencies are given in the table below. Hits 0 1 Frequency 229 211 2 3 4 5 211 93 35 7 1 Assume that the number of hits in a small area (of one-quarter square kilo- meters) has a Poisson distribution with parameter . Note that the Poisson p.m.f, P(Y = y) = = e-11 y! (a) Find the likelihood function for this observed data (you may ignore the terms that are not important for maximum likelihood estimation). (b) Write an R code to plot the likelihood function (for (0.6, 1.2)) (c) Find the maximum likelihood estimate of X (Please do NOT use calcu- lus!) (d) Use the likelihood ratio test to test the null hypothesis Ho : X = 1 against the alternative H : 1