# Median of a Density Function

1. Nov 2, 2008

### ACE_99

1. The problem statement, all variables and given/known data
The probability density function is defined as

f(x) = (4/81)x(9-x^2) for 0 <= x <= 3
= 0 for every other value

Find the median value of the density.

2. Relevant equations
The median of a function is

integral from m to infinity of f(x)dx = 1/2

3. The attempt at a solution

I took the integral from m to 3 of (4/81)x(9-x^2).

This turned out to be

(18/81)x^2 -(1/81) x^4 = 1/2

I then evaluated the left hand side from m to 3 and simplified so that only the variable m is left. The equation I got was

(1/81)m^4 - (18/81)m^2 = -1/2

I know that I am suppose to solve for m in order to get the median but I'm not sure how to do this, and I am also unsure whether my procedure is correct.

Any help would be great thanks. Sorry bout the formatting I'm new to this.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 2, 2008

The integral looks okay. In the equation try setting

$$u = m^2 \Rightarrow u^2 = m^4$$

Then solve the quadratic for $$u$$, and find $$m$$ from that. Remember that $$m$$ has to be between 0 and 3.

3. Nov 2, 2008

### ACE_99

Thanks a lot, I tried that out and it works