Probability theory

a) [10 Marks] Suppose that mutations in a SARS GOV2 (the new Corona virus) occur in the time interval [0, 15] according to the following assump tions: i Divide the interval [0, t] into 72 equal intervals, each of width t ii Let n be very large 1, 0, if 0 mutations occur in the interval 73,7} = 1, 2, . . . ,n if 1 mutation occurs in the interval i,z' = 1, 2, . . . ,n 111 Let Y,- = iv Let P(Y?,- = 1) % At, for large n and for all z' = 1,2,. . . ,n v Let PC\"; > 1) z 0, for large n, for all i = 1,2,...,n vi Let the random variables Y; be independent for z' = 1, 2, . . . ,n for all large n Now let X be the number of mutations that occur in the interval [0,15]. Prove that k )\\t P(X = k) = %, [Hint: Begin by nding a \"starting\" probability model for X that uses the given assumptions. Then let n > 00.] fork=0,1,2,... b) [5 Marks] Let Y1, Y2, . . . ,Yn be n independent Poisson random variables Where E[Y,] = 1 for z' = 1,2,. . . ,n. Show that 5,, = 22.121 Y; has a Poisson distribution and give its expected value and variance