Probability density function of a triangle with unknown height

Question

So I'm wondering how I can come up with a density function for a thing like this. I'm thinking that abs would allow me to ignore the negative side, but I'm not quite certain as to how I should go about building the function when I don't know the height. Does it not even matter in terms of the function?

I also need to find $a$ where $P(X \ge a) = \frac{1}{4}$, but that's hopefully something I can grasp after I've figured out how to build the density function

Answer 1

The probability density must integrate to $1$. The area under the curve is $$ \frac{1}{2}(4)h=1 $$ where $h=\text{height}$. Next to find a function representing the density consider the cases $0\leq x\leq2 $ and $-2\leq x<0$ separately. In each case you are finding the equation of a line joining two points. In the first case, joining $(0,1/2)$ to $(2,0)$ and in the second joining $(0,1/2)$ to $(-2,0)$.

Answer 2

Hint: $$\int_{-2}^{+2} f(x) dx = 1.$$ Also, the area of a triangle with height $h$ and base length of $4$ is $2h$.