# Homework Help: Probability Density Function problem

1. Feb 5, 2017

### Saracen Rue

1. The problem statement, all variables and given/known data
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 $g(x)=f'(x)$.

2. Relevant equations
Knowledge of derivatives, integrals and PDfs.

3. 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.

2. Feb 5, 2017

### andrewkirk

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.

3. Feb 5, 2017

### Saracen Rue

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

4. Feb 5, 2017

### Ray Vickson

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.

5. Feb 5, 2017

### Buzz Bloom

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.
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

Last edited: Feb 5, 2017