Consider a branching process whose family sizes have the geometric mass function f(k)= qpk,k≥ 0, where p + q = 1, and let Zn be the size of the nth generation. Let T = min {n: Zn = 0} bethe extinction time, and suppose that Zo = 1. Find P(T = n). For what values of p is it the case thatE(T) <∞o?

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Answered: Consider a branching process whose…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Yt = 0.6y -1 + 0.2yt-2 + €t, where e is iid N(0, o²). 2.1) Write down the characteristic polynomial equation and find the characteristic roots of this polynomial. Is yt a stationary time series process?
10.14 Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time t = 0), pure water is fed to the tank at a constant rate m(kg/min), and solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed. (a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value. ANSWER + (b) Write a balance on sodium nitrate, letting x(t, m) equal the mass fraction of NaNO3 in the tank and outlet stream. Convert the balance into an equation for dx/dt and provide an initial condition. (c) On a single graph of x versus t, sketch the shapes of the plots you would expect to obtain for m = 50 kg/min, 100 kg/min, and 200 kg/min. (Don't do any calculations.) Explain your reasoning, using the equation of Part b in your explanation. (d) Separate variables and integrate the balance to obtain an expression for x(t, m). Check your solution. Then…
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! It is known that the reliability function for a critical solid state power unit for use 10 ,t20 and t measured in years. 10+t in a communication satellite is R(t) (a) How many units must be placed in parallel in order to achieve a reliability of 0.98 for 5 year operation? (b) If there is an additional common mode with constant failure rate of 0.002 as a result of environmental factors, how many units should be placed in parallel?
Assume the size of a tumort days after a dose of chemotherapy was administered is given by V(t) = 0.332e-0.461 + 0.008e0.461 You are trying to find the optimal time to give the second dose of chemotherapy to minimize discomfort to the patient, while also keeping the tumor in constant decay. When should the second dose be given? a) Between 1 and 3 days after treatment b) Between 9 and 11 days after treatment O c) Between 3 and 5 days after treatment O d) Between 7 and 9 days after treatment
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