a) [10 Marks] Suppose that mutations in a SARS COV-2 (the new Coronavirus) occur in the time interval [0;t] according to the following assumptions:

i Divide the interval [0;t] into n equal intervals, each of width t

ii Let n be very large

iii LetYi=(1;if 1 mutation occurs in the intervali;i= 1;2;:::;n0;if 0 mutations occur in the intervali;i= 1;2;:::;niv LetP(Yi= 1)t=n, for largenand for alli= 1;2;:::;n

v LetP(Yi>1)0, for large n, for alli= 1;2;:::;n

vi Let the random variables Yi be independent for i=1;2;:::;n for all large n

Now let X be the number of mutations that occur in the interval [0;t].Prove that

P(X=k) =(t)ketk!;fork= 0;1;2;:::

[Hint:Begin byfinding a "starting" probability model for Xt hat uses the given assumptions.Then let n!1.]

b)[5 Marks] LetY1;Y2;:::;Yn be n independent Poisson random variables whereE[Yi] = 1 for i= 1;2;:::;n.Show thatSn=Pni=1Yi has a Poisson distribution and give its expected value and variance.

a) [10 Marks] Suppose that mutations in a SARS COV-2 (the new Corona virus) occur in the time interval [0, t] according to the following assump- tions: i Divide the interval [0, t] into n equal intervals, each of width t ii Let n be very large iii Let Yi = if 1 mutation occurs in the interval i, i = 1, 2, . . ., n 10, if 0 mutations occur in the interval i, i = 1, 2, . .., n iv Let P(Yi = 1) ~ At, for large n and for all i = 1, 2, ..., n v Let P(Yi > 1) ~ 0, for large n, for all i = 1, 2, ..., n vi Let the random variables Y; be independent for i = 1, 2, ..., n for all large n Now let X be the number of mutations that occur in the interval [0, t]. Prove that P(X = k) = (At)ke-At k! for k = 0, 1, 2, ... Hint: Begin by finding a "starting" probability model for X that uses the given assumptions. Then let n -> co.] b) [5 Marks] Let Y1, Y2, ..., Yn be n independent Poisson random variables where E[Y ] = 1 for i = 1, 2, ..., n. Show that Sn = Et, Y; has a Poisson distribution and give its expected value and variance