Applied Statistics-Time Series Analysis [ISS (Statistical Services) Statistics Paper III]: Questions 15 - 17 of 17

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Question 15

Describe in Detail Subjective▾

If Xn is a branching process with

and σ2 = Var (X1) , then show that

  1. E (Xn) = mn
Edit

Explanation

Let X0 = 1. It is evident

Let Zij be i. i. d with the offspring distribution P [Zij = k] = pk, k = 0,1, 2, . . Such a process {Xn} is called branching process and Xn denotes the number of individual in the nth g…

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Question 16

Write in Short Short Answer▾

What is Gambler՚s ruin problem? Obtain the probability of ruin, if Gambler plays with a capital of a rupees.

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Question 17

Describe in Detail Subjective▾

Define a Galton- Watson branching process. If {pk} is the offspring distribution, then prove the following identities:

(i) Pn (s) = Pn-1 [P (s) ]

(ii) ) Pn (s) = P [Pn-1 (s) ]

Edit

Explanation

Consider a population consisting of individual, able to produce identical offsprings of the same type and dies after producing. Let Zij be the number of offspring produced by the jth individual of the ith generation and Xn be number of individual of the nth generation, j = 1,2, … , Xi and i = 0,1, 2, …

Let X0 = 1. It is evident

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