Question

3. Assume that X is the number of large earthquakes (with magnitude 2 7.5) occurring in each year. A statistician suggested that X follows a Poisson distribution with parameter ?. A Poisson distribution with parameter ? has expectation ? and variance ?. Suppose a data set 1,22,.,^n is the realization of a random sample Xi,..., Xn from this distribution. One can use either ? 1-X, or ?2-1 ?21 (Xi-%)2 to estimate the parameter ?. (a) Find Eli21 (b) Are both Xi and X2 unbiased estimators of A? (c) The number of large earthquakes (with magnitude 2 7.5) occurred each year (globally) (3 marks) (1 mark) during the years from 2001 to 2010 (x1,^2,... ,1o) is 7, 6, 5, 3, 5, 5, 10, 2, 8, 6 (1 mark) (d) Use the estimate that you obtained in (c) to find P(X > 10). (You can use R to calculate (1 mark) Find an unbiased estimate of A for this sample. this probability.)

Answer 1

a) E(lambda^cap_2) = 1/n sigma^n_i = 1 E(x_i - x^bar_n) ^2 = 1/n{sigma E(x^2_i) - 2 sigma E(x_i) x^bar_n + n x^bar2_n} = 1/n {n (lambda^2 + lambda) - n{V(x^bar_n) + E^2 (x^bar_m) }} = 1/n {n lambda^2 + n lambda - n {lambda/n + lambda^2 }} b) E(lambda^cap_1) = E(x^bar_n) = lambda therefore lambda^cap_1 is use of lambda, lambda^cap_2 is not use of lambda c) lambda = 7 + 6 + 5 + 3 + 5 + 5 + 10 + 2 + 8 + 6/10 = 5.7 d) P(x > 10) = 1 - p(x lessthanareeqaulto 9) = 1- sigma^9_x = 0 e^-5.7 (5.7) ^x/x! = 0.0648175