solve this

1. Consider a random variable X whose density function is dBen4 fx(z) = ao za=0,1,2,... i) What is the name of this common distribution? ii) The blog \"Earth changes and pole shift\" (at http: //poleshift .ning. com) suggests that there were a remarkably high number of major earthquakes at the start of this year. There were four major earthquakes in just the first 18 days of 2011, which is higher than the long-term average of 16.25 major earthquakes per year. a) Suggest a suitable value of A that could be used to model X, the number of major earthquakes in the first 18 days of 2011 (given that we expect 16.25 earthquakes per year). b) Calculate (to three decimal places) a probability that measures how unusual it is to have four or more earthquakes in the first 18 days of the year. it Show that the moment generating function of X is: mz(u) = Me il Hence or otherwise find the distribution of X,+-X where X, and X are independent observations, both of which have density function fx (xr). v) A random variable is said to be infinitely divisible if for any positive integer nm, it can be written as a sum of n identically and independently distributed (iid) random variables X1,Xo,...,Xn- i Is X infinitely divisible? If so, state the distribution of the iid X; such