ii] [12 marks] For some J. 3:- 0, let X v-u Poisson}, so the probability function is given by a} b} d} Are1 ml fx{r) = Pr{X = r) = , r=,1.2,... Show that 23:.) fx{r] = 1. For this part of the question1 assume that the earth has experienced a long-term average of 16.25 major earthquake; per year. The Poisson random variable can be used to model the number of major earthquakes in a xed period of time. I Suggest- a suitable value of .1 that could be used to model the number of major earthquakes in a random month {a month is of a year]. I Using your suggested A, calculate {to three decimal places] the probability that no earthquakes occur in a random month. II There 1were three major earthquakes in just the rst month of 2013. Using your suggested Jr, calculate {to three decimal places] a probability that measures how unusual it is to have three or more earthquakes in a random month. In plain language, com~ ment on whether you think this probability is unusual. Show that the moment generating function of X is: ma) : assu. Let X11X3,X3, .. x]: be independent Poissonfh] random variables. Set W = X1+X2+X3+- \"+Xk. Using the moment generating flmc~ tion of X [or otherwise] determine the moment generating function for W