Probability Density Function problem

[Saracen Rue]

Homework Statement

Q6. A function, $f\left(x\right)=\frac{ax+1}{\left(ax-1\right)^3-\frac{a}{\left(x-1\right)^2-1}}$, can be defined as a PDf over the domain $(0, 2)$.

Express answers to 4 decimal places unless specified otherwise;
(a) Find the value of $a$ given that $f(x)$ is a PDf
(b) Calculate the mean, variance and standard deviation correct to 3 significant figures.
(c) Determine, correct to one decimal place, the percentage probability of discrete random variable $x$ being within two standard deviations either side of the mean.
(d) State the value of the median, $m$
(e) Find the maximal value of $f(x)$ and determine the percentile of discrete random variable $x$ at this point.
(f) The derivative of function $f(x)$ can also be defined as a PDf over the domain $(0, c]$. Find the value of $c$ correct to 3 decimal places, letting $g(x)=f'(x)$.

Homework Equations

Knowledge of derivatives, integrals and PDfs.

The Attempt at a Solution

The first thing I did was simplify $f(x)$ as I believed it looked rather messy and hard to work with. Sadly I didn't get too far with this - the best I could do was to express it as $f(x)=\frac{x\left(ax+1\right)\left(x-2\right)}{a\left(3x^3-6x^2-1\right)+\left(x-2\right)\left(a^3x^4-3a^2x^3-x\right)}$

I then promptly became stuck at the first question. I understand that the question wants me to integrate $f(x)$ using $0$ as the lower limit and $2$ as the upper limit, set whatever my answer is to equal $1$, and then solve for $a$. However, I'm not sure how to integrate the function - I don't even think there is a way to calculate the definite integral of this function. Normally this wouldn't be a problem because I'd just use my calculator to integrate, but it's having trouble doing it with two unknowns present. I could substitute in random values of $a$ and then integrate until I get close to the integral equaling $1$, but that seems like it would be an excessive amount of work and would take a lot of time. Is there another method I could use here to get past this first question? Thank you all for your time and help.

 

[andrewkirk]

The first thing I did was simplify $f(x)$ as I believed it looked rather messy and hard to work with. Sadly I didn't get too far with this - the best I could do was to express it as $f(x)=\frac{x\left(ax+1\right)\left(x-2\right)}{a\left(3x^3-6x^2-1\right)+\left(x-2\right)\left(a^3x^4-3a^2x^3-x\right)}$

I then promptly became stuck at the first question. I understand that the question wants me to integrate $f(x)$ using $0$ as the lower limit and $2$ as the upper limit, set whatever my answer is to equal $1$, and then solve for $a$. However, I'm not sure how to integrate the function - I don't even think there is a way to calculate the definite integral of this function.

The function is a rational function - meaning a ratio of polynomial functions. Any rational function can be integrated using partial fractions to decompose it into more manageable pieces. For instance see here.

There may be a neater, easier way to do it, as the partial fractions approach will be somewhat long. But a more elegant approach is not leaping out at me right now.

 

[Saracen Rue]

The function is a rational function - meaning a ratio of polynomial functions. Any rational function can be integrated using partial fractions to decompose it into more manageable pieces. For instance see here.

There may be a neater, easier way to do it, as the partial fractions approach will be somewhat long. But a more elegant approach is not leaping out at me right now.

Thanks for the advice!

I'll read up on on partial fraction integration via the link you provided and then give the question another shot.

 

[Ray Vickson]

Thanks for the advice!

I'll read up on on partial fraction integration via the link you provided and then give the question another shot.

I don't think the partial fraction approach will work in practice; the denominator is a 5th degree polynomial containing $a$ is some of its coefficients, so the roots of the denominator will be functions of $a$---possibly being unobtainable in closed-form (because it is known to be impossible to give a nice formula for roots of a polynomial of degree >= 5). However, just computing the integral numerically for various numerical values of $a$ works well, especially if you use a computer package such as Maple or Mathematica.

 

[Buzz Bloom]

the best I could do was to express it as f(x)=

Hi Saracen Rue:
I agree with Ray that numerical integration with various trial values for a is the best approach. Here is an online integration site I have used hat might be helpful.

http://solvemymath.com/online_math_calculator/calculus/definite_integral/index.php​

You may first want to find two values of a such that one gives an integral greater than 1, and the other less than 1. Then keep dividing the reduced range in half. However that may take perhaps 10-20 tries to achieve 4 digits of precision, so some patience will be needed.

After you have found a value for a that is OK, you can integrate x f(x) and x² f(x) to calculate mean, variance, and standard deviation.

Regards,
Buzz