Question

It is felt that error in measurement of reaction temperature (in celsius) in an experiment is a continuous r.v. with p.d.f.

f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}`
Verify whether f(x) is a p.d.f.

Answer 1

Here, f(x) ≥ 0, x ∈ [0, 4]

Now consider,

`int_0^4 f(x).dx`

= `int_0^4 x^3/(64).dx`

= `(1)/(64) int_0^4 x^3.dx`

= `(1)/(256)[x^4]_0^4`

= `(1)/(256)[256 - 0]`

= 1

∴ f(x) = `{(x^3/(64),  "for"  0 ≤ x ≤ 4),(0,   "otherwise."):}` is a p.d.f