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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.
Solution
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
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