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प्रश्न
Check whether the following is a p.d.f.
f(x) = `{(x, "for" 0 ≤ x ≤ 1),(2 - x, "for" 1 < x ≤ 2.):}`
उत्तर
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
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