- #1

Saracen Rue

- 150

- 10

## Homework Statement

A function, ##f\left(x\right)=\left|a^{\frac{\sin \left(x\right)}{\ln \left(ax\right)}}-\frac{x}{a}\right|##, intersects with another function, ##g\left(x\right)=\left|\frac{sin(ln(\sqrt{x}-\sqrt{a}))}{x^2-a^2}\right|##, at point ##Q(b,f(b))## and point ##R(c,f(c))##. A probability density function, ##p(x)=g(x)-f(x)##, can be formed over the domain ##[b,c]##, where ##0<b<c<2##.

**Q1.**Determine, correct to 16 decimal places;

**(a)**The value of the real-valued constant, ##a##

**(b)**The coordinates of the points ##Q## and ##R##

**Q2.**Calculate the mean, variance and standard deviation correct to 4 decimal places

**Q3.**Find correct to 1 decimal place; the percentage probability of the continuous random variable, ##X##, being within ##a^2## variances either side of the mean (i.e. ##Pr(μ-a^2⋅Var(X)≤X≤μ+a^2⋅Var(X))##)

## Homework Equations

For PDfs: ##∫_{b}^{c}f(x)dx=1##

## The Attempt at a Solution

I know that to find points ##b## and ##c## I need to solve ##f(x)=g(x)## for ##x##, however when I try to do this on my calculator it gives me an error message. I know that I should be getting ##b## and ##c## in terms of ##a##, thus ##∫_{b}^{c}p(x)dx=1## should have ##a## as the only unknown and it should be able to be solved for. After getting ##a##, I'd substitute the value into ##b## and ##c## to get those values as well, and then I'd be able to find ##f(b)## and ##f(c)## as well (giving me points ##Q## and ##R##).

To find the mean, variance and standard deviation I'd just solve the following:

##E(X)=∫_{b}^{c}x⋅p(x)dx##, ##Var(X)=∫_{b}^{c}x^2⋅p(x)dx## and ##σ=\sqrt{Var(X)}##

And for ##Q3## I'd solve this: ##Pr(μ-a^2⋅Var(X)≤X≤μ+a^2⋅Var(X))=∫_{μ-a^2⋅Var(X)}^{μ+a^2⋅Var(X)}p(x)dx##

I am fairly confident with my method for the most part (although I'd still appreciate somebody telling me if there's any errors in it), the main issue I'm having is simply not being able to solve the equations on my calculator. I'd be very thankful for anybody who has an alternative way for me to solve this problem :)