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Question
Verify which of the following is p.d.f. of r.v. X:
f(x) = sin x, for 0 ≤ x ≤ `π/2`
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Solution
(a) f(x) = sin x ≥ 0 if 0 ≤ x ≤ `π/2`
(b) `int_(-∞)^∞ f(x) dx = int_(-∞)^0 f(x) dx + int_(-∞)^(π/2) f(x) dx + int_( π/2)^∞ f(x) dx`
= `0 +int_(0)^(π/2) sinx dx + 0`
= `[-cosx]_0^(π/2) = -cos(π/2) + cos0 = 0 + 1 = 1`
Hence, f(x) is the p.d.f. of X.
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