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Question
Choose the correct option from the given alternative:
P.d.f. of a.c.r.v X is f (x) = 6x (1 − x), for 0 ≤ x ≤ 1 and = 0, otherwise (elsewhere)
If P (X < a) = P (X > a), then a = .....
Options
1
`1/2`
`1/3`
`1/4`
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Solution
`1/2`
Explanation:
Given:
f(x) = 6x(1 − x), 0 ≤ x ≤ 1
P(X < a) = P(X > a)
Since total probability = 1,
P(X < a) = P(X > a) `= 1/2`
`int_0^a 6x(1-x) dx = 1/2`
Integrate
`int 6x(1-x) dx = int (6x-6x^2)dx`
= 3x2 − 2x3
`3a^2 - 2a^3 = 1/2`
`2a^3 - 3a^2 + 1/2 = 0`
4a3 − 6a2 + 1 = 0
`4(1/8)-6(1/4) + 1 = 0`
`=1/2 - 3/2 + 1 = 0`
`a = 1/2`
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