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`

Answer 1

`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`