Recent content by Kvad

Thank you, it makes sense now!

Hello, I have tried the integral below with Mathematica and it gives me the following solution: ##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}## I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...

Well unfortunately I am interested only in the general case... But maybe if I rewrite ##∫F(x)f'(x)dx## as ##∫F(x)df(x)## getting an integration of CDF by PDF, maybe this problem is known to have some solution or on the contrary unsolvable? I tried to substitute ##f(x)=z## and then get an easy...

Something that will not have any integrals and derivatives, expressed in parameters and f(x) and F(x)...

I am looking just for an analytical answer, If I do integration by parts correctly then I get this? ## F(x)f(x) = ∫f'(x)F(x)dx + ∫f(x)^2dx ## Now ##∫f(x)^2dx=F(x)f(x)-∫F(x)f'(x)dx## and I am not sure about how to tackle the last integral?

Hi all, I was trying to find an answer, but couldn't, what is the integral of the squared probability density function? It doesn't seem to be equal to the square of cumulative distribution function, but how to tackle it? ∫(f(x))2dx = ? Can we transform it into, say, ∫f(x)dF(x)? and then...

Hi all, I have derived a differential equation, which I don't know how to solve. I can do some numerical simulations, but would really be interested in, at least, knowing if an analytical solution exists, so would appreciate any help with it: (I have removed argument from y)...

Thank you so much, Simon, for all your suggestions and hints! I really appreciate it, but unfortunately I cannot say more about the problem beyond its mathematical formulation...

I am sorry but I did not understand the question about "what level am I doing it at", could you, please, specify? As for the general form you are right, f(x)+g(y)=h(y)y' is the one! I do not know much about differential equations, could you, please, refer me to a method, which deals with such...

Yes, and the equation is more or less solvable when n=3, but starting from n=4 there is no more difference of squares which allowed for simplification... Say, for n=5 we have: $$\frac{x^3(3x-4p)+4py^3-3y^4}{y'(x)}-6y^2(p-y)^2=0$$ or the general form of the same equation...

Hi guys! I am really stuck at quite a complicated (from my point of view) differential equation. I would really appreciate any hints or suggestions on how to tackle and solve it if it is possible, thanks...

Ok, thanks again for all your answers!

Yes, this I understand, but I wonder does the analytical way you solved this equation allows to come to some kind of explicit solution that would map x to y, so that when, say, p=0,5 or any other number withing given range, we would always be able to identify the value of function y, given the...

Ok, thanks again! One last naive question, could you, please, tell me if the analytical solution also shows the downward slope of the function, because I am not sure I completely understand it...

Do you mean the initial equation, where we are given that basically y<p<x and therefore the slope is evidently negative, or do we know for sure the signs of X(\theta) \ \ and \ \ \theta ? In this case from the parametric solution I believe it would be a more formal proof...