Newton-Raphson method in non-homogeneous poisson process

[tottijohn]

Homework Statement

The rate of occurrence of events in a non-homogeneous Poisson process is given by: λ(t)=12t e^-2t.

(c) Find the p.d.f. of the time until the first event occurs after time t = 1.
(e) After what time is it 95% certain that no further events will occur?

Homework Equations

λ(t)=12t e^-2t

The Attempt at a Solution

After using integration by parts, I found μ(t) = -3e^-2t(2t+1) to solve other parts of this question. I know part (e) requires the use of Newton-raphson method but I have no idea how to go about. Any help will really be appreciated, thanks.

 

[hotvette]

Newton-Raphson is a numerical method to estimate the zero of a function (i.e. find x such that f(x) = 0) to some desired accuracy. Are you having difficulty defining f(x) or applying NR or both?

 

[tottijohn]

Both.

I know the probability of extinction by each generation can be calculated using Newton_Raphson method given p.g.f. and finding p and q, but I am not sure how to apply to this question.

 

[Ray Vickson]

Both.

I know the probability of extinction by each generation can be calculated using Newton_Raphson method given p.g.f. and finding p and q, but I am not sure how to apply to this question.

Are you trying to solve an equation? What IS the equation? Write it down in detail, so we have the basis for offering some advice.

RGV

 

[tottijohn]

I have managed to solve (c).

Here is the equation i got for (e):
g(t) = (3 + 6t)e^-2t - 0.0513 = 0
g'(t) = -12te^(-2t)
t_n+1 = t_n - [(3 + 6t_n)e^-2t_n - 0.0513]/-12t_ne^(-2t_n)

I am not sure how to proceed with NR to get the time of 95% certain?

 

[Ray Vickson]

I have managed to solve (c).

Here is the equation i got for (e):
g(t) = (3 + 6t)e^-2t - 0.0513 = 0
g'(t) = -12te^(-2t)
t_n+1 = t_n - [(3 + 6t_n)e^-2t_n - 0.0513]/-12t_ne^(-2t_n)

I am not sure how to proceed with NR to get the time of 95% certain?

Why the concentration on Newton-Raphson? Do you understand that you are just trying to solve the equation (3 + 6t)*exp(-2t) = 0.051293? Newton-Raphson (NR) is one way to do it, but there are many others. However, if you do want to use NR to solve the equation g(t) = 0, you just start with some initial guess, t₀, then use the iteration scheme
t_n+1 = t_n - g(t_n)/g'(t_n). What is stopping you from doing this?

RGV