2

In order to prove the following Probability result:

"Find a family of random variables $X$, having pdf $f$, such that $X$ and $Y=f(X)$ have the same distribution."

Let's call $F(x)$ the CDF of the random variable $X$ and $f(x)$ its PDF. Furthermore, the random variable $Y$ has CDF $K(y)$ and PDF $k(y)$.

I know that $y$ belongs to a set of values in which it can be considered valid the relation $y=f(x)$.

I'd like to "investigate" when the random variables $X$ and $Y$ have the same distribution, in the case of $f$ stictly increasing for $x\leq m$ and stricly decreasing for $x \geq m$ (where $m$ is the mode of the distribution).

So we have to prove when $F(y)=K(y)$.

From a previous valid result, I know that, under these assumptions, the CDF of the random variable $Y$ is:

$K(y)=2 F(f^{-1}(y))=2F(x)$ or, equivalently, $K(y)=2(1-F(f^{-1}(y))=2-2F(x)$.

In order to find the explicit expression of the PDF $f$, using the relation $F(y)=K(y)$ (our thesis), I should prove that:

  1. in the case $f$ strictly increasing for $x \leq m$,

$$K(y)=2F(x)=F(y) \Leftrightarrow 2F(x)=F(f(x)) \Leftrightarrow 2F(x)=F(F'(x)).$$

So the first functional differential equation that should be solved is:

$$2F(x)=F(F'(x))$$ in order to find the expression of $F$ and then obtain $f$, derivating $F$.

  1. in the case of $f$ strictly decreasing for $x \geq m$

$K(y)=2-F(x)=F(y) \Leftrightarrow 2-F(x)=F(f(x)) \Leftrightarrow 2-F(x)=F(F'(x))$.

The second functional differential equation to be solved is the following one: $$F(F'(x))+F(x)=2.$$

The solutions of the aforesaid equations should be in a certain way "symmetrical".

I should solve those functional differential equations, but I have NO IDEA on HOW to solve them because I've never met this kind of equations before in my career.

Could you help me, please?

met.91
  • 97
  • Typically I've seen functional diff eq's approached by Laplace transforms. Unlike ODE's, things get interesting: infinite polynomials in 1/s result, which can sometimes be solved. What happens in this case? See the papers by Fox " On a functional differential equation..." in the early 70s. – kevinkayaks Jun 18 '21 at 21:59
  • 1
    @met.91 Can you elaborate on how you got the two FDEs? – Mohsen Shahriari Jun 19 '21 at 00:42
  • Related to other questions you have asked, including https://math.stackexchange.com/questions/4168489/find-a-family-of-random-variables-x-with-density-f-such-that-x-and-y-f – Henry Jun 19 '21 at 00:45
  • @met.91 You've already asked a similar question here which has been closed. Please avoid asking duplicates, and try to improve the original post so that it gets reopened. – Mohsen Shahriari Jun 19 '21 at 00:46
  • @Mohsen Shahriari I've already asked the same question, but it was closed (quite instantly) by some admin, I suppose. I thought that it wasn't visible anymore. Maybe I'll improve that and I'll avoid to duplicate the questions (even if people was not able to answer to the other question anymore). I'm new on this site and I'm still understanding how this works. – met.91 Jun 19 '21 at 11:05
  • @met.91 While there are few moderators who possess some extra powers due to extra responsibilities, almost everything on the site is managed by the community itself. Your previous question was closed by five votes from five different users, because in their eyes, it fails to meet the required standards (see How to ask a good question.). Your post is not deleted, so it is still visible. If you improve it by editing it, people may vote to reopen it, and then answers can be posted. – Mohsen Shahriari Jun 19 '21 at 23:08
  • 1
    @met.91 For now, what I suggest is adding your work to the current post, so that this becomes of higher quality, and doesn't end up getting closed like the other one. Show how you got from the original question to the two FDEs for example. Be accurate about thing you're writing. For example, what is the relation between "$ F $" and "$ f $"? Is $ F $ the CDF corresponding to the PDF $ f $? Should the FDEs hold on the whole real line, or one is for $ x \le m $ and the other for $ x \ge m $? Also, add some context. For example, why is the question interesting, and where does it happen to occur? – Mohsen Shahriari Jun 19 '21 at 23:18
  • 1
    @met.91 I also suggest you delete the other duplicate post. In case you do, I think the useful comment by karakfa (which has two upvotes at the moment) should not be lost. It reads: "For the first one you can try $ a x ^ 2 $ and solve for $ a $.". – Mohsen Shahriari Jun 19 '21 at 23:23
  • @MohsenShahriari I've just updated the post, adding more information on HOW I elaborated the equations, which are the goals of my work and why it's important to me to know the solutions of the aforesaid FDEs. – met.91 Jun 20 '21 at 08:51

1 Answers1

1

I tried to solve for $a$ the first FDE

$$F(F'(x))=2F(x)$$

after putting $F(x)=ax^{2}$ and $F'(x)=2ax$ (as suggested by karakfa).

I obtained the equation

$$2x^{2}a(2a^{2}-1)=0$$

and the results are $a=0; \frac{\sqrt{2}}{2}; -\frac{\sqrt{2}}{2}$.

I've chosen $a=\frac{\sqrt{2}}{2}$ so that

$$F(x)=\frac{\sqrt{2}}{2}x^{2}$$

and

$$F'(x)=f(x)=\sqrt{2}x.$$

So the PDF of the random variable $X$ could be the following one:

$$f(x)= \begin{cases} \sqrt{2}x, \, \text{$x \in [0, 2^{1/4}]$} \\ -\sqrt{2}x, \, \text{$x \in [-2^{1/4}, 0)$} \end{cases} $$

but I don't know if this could make any sense.

met.91
  • 97