How to find a probability density function please?

I have the following system :

dxdt = A.*x.^2 ;

how can I find a PDF for this system for different values of A and t = 0:0.1:100?

Regards

Answer 1

[Torsten]

What do you want to be stochastic in the above ODE ?

Best wishes

Torsten.

[Avan Al-Saffar]

The parameter A to be A0*cos(t) , I want to find pdf for different A0.

Can you help me please?

Regards

[Torsten]

The pdf of what ? The pdf of the solution at a fixed time t ?

Best wishes

Torsten.

[Avan Al-Saffar]

Yes, please.

Regards

[Torsten]

Solve the ODE analytically, fix x(t=0) and t, generate random numbers for A0 and insert the random numbers in the analytical solution. This gives you a vector of random numbers for the result of your ODE at time t. Use hist to make a normed histogram of this solution vector to get the pdf.

Best wishes

Torsten.

[Avan Al-Saffar]

Many thanks for your help. Can you also inform me how to do it if I need to find pdf at different time please?

Regards Avan

[Torsten]

Yes, loop over different values for t and insert these values in the analytical solution. This will give you multiple result vectors, multiple histograms and thus multiple pdfs.

Best wishes

Torsten.

[Avan Al-Saffar]

Many thanks.

Regards

Avan

[Avan Al-Saffar]

But what is about the following commands please?

- ksdensity(x)

- [muhat,sigmahat] = normfit(x);

                   y = normpdf(x,muhat,sigmahat);

                                      plot(x,y,'r')

- a = pdf('Normal',x,mu,sigma);

I found them while I was looking for some help through different websites.

[Torsten]

You can use ksdensity, but if you use enough random numbers for A0, a histogram will give you the same quality for the pdf.

Normfit, normpdf etc. can not be used because even if you assume that A0 is normally distributed, x(t,A0) usually won't.

You will have to find out what distribuion x(t,A0) follows if you assume a certain distribution type for A0.

Best wishes

Torsten.