# A PDF formed from a multivariate function

1. Oct 2, 2016

### Saracen Rue

1. The problem statement, all variables and given/known data
A probability density function, $p\left(x\right)=\int _{-a}^a\left(f\left(x,\left(a-y\right)\right)-f\left(y,ax\right)\right)dy$, can be formed from the bivariate function $f\left(x,y\right)=y\sqrt{\left(x-2\right)^2-y}$ over the domain $[0,b]$ - where $b$ is the coordinate of the x-intercept.

(a) Determine the values of $a$ and $b$ correct to 5 decimal places
(b) Calculate the expected value, variance and standard deviation of $p(x)$ correct to 4 decimal places
(c) Find the percentage probability of the continuous random variable $X$ being withing $|a|$ standard deviations either side of the mean​

2. Relevant equations
Knowledge or probability density functions, including integral applications.

3. The attempt at a solution
Well I know that I need to substitute the actual values for $f\left(x,\left(a-y\right)\right)$ and $f\left(y,ax\right)$ into $p(x)$ to be able to integrate $\int _{-a}^a\left(f\left(x,\left(a-y\right)\right)-f\left(y,ax\right)\right)dy$, and then I'd need to solve ##int_0^b p(x)dx for b and then use that to solve for a. However, when I attempt to do this on my calculator I receive and error message. Does anyone know how I could do this?

2. Oct 7, 2016

### Staff: Mentor

Which error message do you get?

3. Oct 9, 2016

### Saracen Rue

I receive the message: "ERROR: Insufficient Memory"

4. Oct 10, 2016

### Staff: Mentor

Maybe you can simplify the function a bit, or use a better calculator.