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प्रश्न
Verify which of the following is p.d.f. of r.v. X:
f(x) = 2, for 0 ≤ x ≤ 1.
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उत्तर
f (x) is the p.d.f. of r.v. X if
(a) f (x) ≥ 0 for all x ∈ R and
(b) `int_(- ∞)^∞ f(x) dx = 1`
(a) f(x) = 2 ≥ 0 for 0 ≤ x ≤ 1
`int_(- ∞)^∞ f(x) dx = int_(- ∞)^0 f(x) dx + int_(0)^1 f(x) dx +int_( 1)^∞ f(x) dx`
= 0 +`int_0^1 2dx`+ 0
= [2x]01
= 2 - 0
= 2 ≠ 1
Hence, f (x) is not p.d.f. of X.
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