Please help me in solving this? I need it urgently to submit my assignment pleas

Question

Please help me in solving this? I need it urgently to submit my assignment please.

3. In a branching process, the number of offspring per individual has a Binomial(2, p) distribution, where p> 1/2. Starting with a single individual a. Show that the extinction probability ° is equal to 921@, where q = 1-p. b. If p = 3/4· find the probability that the population becomes extinct for the first time at the 3rd generation. (Hint: this is NOT equal to P(X3 = 0), because the event AG = 0} includes outcomes for which the population is 0 in the 2nd or 1st generations.)

Explanation / Answer

a) The probability for binomaial is as follows P(0)=(1-p)^2, P(1)=2p(1-p) and P(2)=p^2,

Thus P(x=1 or 2)=2p(1-p)+p*2=1-p^2.

Thus extinction probability is P(0)+P(1,2)*P(0)+P(1,2)^2*P(0).... which is

(1-p)^2*[1+(1-p^2)+(1-p^2)^2+...]=(1-p)^2/(1-(1-p^2))=(1-p)^2/p^2 or q^2/p^2

b)P(x3=0) is P(1,2)*P(1,2)*P(0)=(1-p^2)^2*(1-p)^2 for p=3/4 it is (1-9/16)^2*1/4*1/4=49/256*1/16=49/4096=.0119

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