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The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t≥ 0}. A partial differential equation for П(s, t), the probability generating function of X(t), is ап Ət = -ßs(1 - s) With general solution ап მა II(s, t) = ¢ (₁²,e-²). Suppose that X(0), the number of individuals alive at time 0, is a random variable: X(0) has the negative binomial distribution with range {4, 5, ...} and parameters r = 4 and p=0.8. Find the particular solution corresponding to this initial condition. Hence identify the probability distribution of X(t) in this case, and find its mean

The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t≥ 0}. A partial differential equation for П(s, t), the probability generating function of X(t), is ап Ət = -ßs(1 - s) With general solution ап მა II(s, t) = ¢ (₁²,e-²). Suppose that X(0), the number of individuals alive at time 0, is a random variable: X(0) has the negative binomial distribution with range {4, 5, ...} and parameters r = 4 and p=0.8. Find the particular solution corresponding to this initial condition. Hence identify the probability distribution of X(t) in this case, and find its mean

A First Course in Probability (10th Edition)
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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The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t≥ 0}. A partial differential equation for П(s, t), the probability generating function of X(t), is ап Ət -Bs(1s) With general solution ап Əs II(s, t): = 4 (₁ ²²₂e²²²). Suppose that X(0), the number of individuals alive at time 0, is a random variable: X(0) has the negative binomial distribution with range {4, 5, ...} and parameters r = 4 and p = 0.8. Find the particular solution corresponding to this initial condition. Hence identify the probability distribution of X(t) in this case, and find its mean
Transcribed Image Text:The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t≥ 0}. A partial differential equation for П(s, t), the probability generating function of X(t), is ап Ət -Bs(1s) With general solution ап Əs II(s, t): = 4 (₁ ²²₂e²²²). Suppose that X(0), the number of individuals alive at time 0, is a random variable: X(0) has the negative binomial distribution with range {4, 5, ...} and parameters r = 4 and p = 0.8. Find the particular solution corresponding to this initial condition. Hence identify the probability distribution of X(t) in this case, and find its mean
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