Question

Problem 2. A random variable has the following density function:

Answer 1

If f(x) is a density function, then its integral over the entire real line should be 1:

$\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=\int_0^2(1-0.5x)\,\mathrm dx=(x-0.25x^2)\bigg|_0^2=2-1=1$

so yes, f(x) is indeed a density function.