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Question
The p.d.f. of a continuous r.v. X is given by
f (x) = `1/ (2a)` , for 0 < x < 2a and = 0, otherwise. Show that `P [X < a/ 2] = P [X >( 3a)/ 2]` .
Solution
P (`X< a /2`) = ` int_(-∞)^(a/2) f (x) dx`
=` int_(-∞)^0 f (x) dx+ int_(-∞)^(a/2) f (x) dx`
= 0+ ` int_(0)^(a/2) 1/(2a) dx`
= `1/(2a) int_(0)^(a/2) 1 dx = 1/(2a)[x]_0^(a/2)`
= `1/(2a) [a/2-0] = 1/4` .......(1)
P `[X >( 3a)/ 2] = int_((3a)/2)^(∞) f(x) dx`
= `int_((3a)/2)^(*2a) f(x) dx + int_(2a)^(∞) f(x) dx`
= `int_((3a)/2)^(2a) 1/(2a) dx + 0`
= `1/(2a) int_((3a)/2)^(2a) 1 dx = 1/(2a)[x]_((3a)/2)^(2a)`
= `1/(2a)[2a-(3a)/2] = 1/(2a)(a/2) = 1/4` ......(2)
From (1) and (2), we get
`P [X < a/ 2] = P [X >( 3a)/ 2]`.
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