Probability Density Function problem

[Saracen Rue]

Homework Statement

Presume the relation $\frac{x}{x+y^2}-y=x$ is defined over the domain $[0,1]$.

(a) Rearrange this relation for $y$ and define it as a function, $f(x)$.
(b) Function $f(x)$ is dilated by a factor of $a$ from the y-axis, transforming it into a probability density function, $p(x)$. Find the value of $a$ correct to 4 decimal places.
(c) Determine the following correct to 3 decimal places:

I) The mean of $p(x)$
II) The standard deviation of $p(x)$
III) The median, $m$, of $p(x)$​

(d) Calculate the probability of discrete random variable $x$ being within $a$ standard deviations either side of the mean.

Homework Equations

Knowledge of integration, probability density functions, and the rearranging and solving of equations.

The Attempt at a Solution

Starting with part $(a)$, I attempted to rearrange $\frac{x}{x+y^2}-y=x$ for $y$. I managed to express the equation in the form $y^3+xy^2+xy+x^2-x=0$ however this is where I become stuck. I'm unsure of how to factorise this equation for $y$ and my calculator simply returns an error message when I try and use it. Is there another way to do this that I'm missing or don't know about?

Thank you for taking your time to read this :)

 

[andrewkirk]

Your calculations so far look correct. I don't think that formula can be factorised. A formula for the solution of the cubic in y could be written (see cubic solution formula here) but it would be very messy and I doubt that's what was intended.
Perhaps the question contains an error like a wrong sign, and the question it was supposed to be is factorisable where the one they actually wrote is not.

Where did you get the question? It has a number of other errors, such as

• the statement that the relation is over the domain [0,1], which cannot be correct as there are two variables and [0,1] is only one-dimensional. Perhaps they meant to say [0,1] x [0,1].
• the references to the mean, standard deviation and median of p(x) are meaningless, since those statistics are properties of random variables and p(x) is not a random variable. Perhaps they meant to say the random variable whose pdf is p.

 

[Charles Link]

There may be a simple way to factor it, but with a little effort in trying to factor it, I came up empty. Suggestion would be to do the formal solution of the cubic equation on it in order to determine the factors.

 

[Charles Link]

A follow-on: I worked through most of the formal solution of the cubic equation for this problem. Unless I made algebraic errors, it doesn't appear to simplify a great deal and you get complicated polynomials of powers of x from the 5th power to the second power (the 6th power term cancelled) inside of a square root sign. That part is the solution to a quadratic equation that you add an expression consisting of a 3rd power polynomial in x and then take a cube root of it. Finally, you would then do a similar computation to get the "s" term, (solving for s and t), y'=s-t, and then y=y'-x/3. This one does not appear to be simple.

 

[haruspex]

Maybe you are not expected to get it into the form y=... Try proceeding to the next part, which involves computing ∫y.dx. Can you see a way to do that?