- #1

Saracen Rue

- 150

- 10

## 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.