Question

Check whether the following is a p.d.f. 

f(x) = `{(x, "for"  0 ≤ x ≤ 1),(2 - x, "for"  1 < x ≤ 2.):}`

Answer 1

Here, f(x) ≥ 0 `AA` x ∈[0, 2]
Also, f(x) is continuous.
Now consider,

`int_0^2 f(x)*dx = int_0^1f(x)*dx + int_1^2f(x)*dx`

= `int_0^1x*dx + int_1^2(2 - x)*dx`

= `(1)/(2)[x^2]_0^1 + 2[x]_1^2 - (1)/(2)[x^2]_1^2`

= `(1)/(2)[1 - 0] + 2[2 - 1] - (1)/(2)[4 - 1]`

= `(1)/(2) + 2 - (3)/(2)`
= 1
∴ f(x) is p.d.f. of r.v.X