Question

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

f (x) = 1- 0.5x 0 < x < 2


a. Graph the density function.
b. Verify that f(x) is a 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.