Thus, the proportion of time is restricted to:
[tex]\pi = 0.208[/tex]
Given that an exponential rate [tex]\lambda[/tex], at an expoential rate [tex]\mu[/tex]. An exponential rate of increase [tex]\theta[/tex] : the popuilation size is N.
Hence the total birth rate where there are n persons in the system is: [tex]n\lambda+\theta[/tex]. are assume to accury at an expoential rate [tex]\mu[/tex] for each member of the population. So: [tex]n\mu = \mu n[/tex]
Let x(t) denote the population size at time t. Suppose: [tex]x(0)= ix(0)= i[/tex] and let [tex]M(t)= E[x(t)][/tex]. So they will determine M(t) by deriving and then solving a differential equation that is :
We start by deriving on equation for M(t+h) by conditioning on X(t) this yelds:
[tex]M(t+h)= E[x(t+h)]= E(E[x(t+h)] x(t))[/tex].
Now, given the size of the population at the time then, ignoring events whose probability is [tex]\theta (h)[/tex]. The population at time t+h. Will be either increase in size by 1. If a birth or immigration occurs in (t, t+h) or decrease by 1 if a death occurs in this internal, or remain the same if neither of these two possibilities occurs that is given [tex]x(t)*x(t)[/tex]:
[tex]x(t+h)x(t+h)=x(t)+1[/tex] with pobability
[tex][\theta +x(t)x]h +\theta h \\[/tex]
X(t)-1 with probability [tex]x(t)\mu h +\theta (h)[/tex]
x(t) with probability [tex]1-[\theta + x(t) \lambda + x(t) \mu ] h + \theta(h)[/tex]
[tex]E[x(t+h)x(t)]= x(t)+[\theta+ x(t)\lambda-x(t)\mu]h+\theta(h)[/tex]
Since the probabilities have to add up to 1, that determines. The value of the probabiliy measure for he best case.
For [tex]0\leq i\leq N[/tex]
we have [tex]\lambda i = i \lambda + \theta[/tex]
For [tex]i \geq N[/tex] we have [tex]\mu i = i \mu[/tex]
[tex]\pi[/tex] has [tex]\pi \theta = \frac{1}{1+ \sum \phi n }[/tex]
where [tex]\phi n = \frac{\lambda}{\mu}[/tex]
When [tex]\theta=\lambda[/tex]
For [tex]n\geq 3[/tex] we see [tex]\phi n + \frac{1}{\phi n } = \frac{n \lambda}{(n+\mu) \mu}[/tex]
This leads to [tex]\pi = 0.208[/tex]
Learn more: brainly.com/question/17001402
