If a continuous random variable x has the probability density function

\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3x^2,}&o\le x\le1\\ {0,}&{elsewhere} \end{array}} \right.\)

then the value of a such that P[x ≤ a] = P[x > a] is:

Question

Question

This question was previously asked in

SSC CGL Tier-II ( JSO ) 2016 Official Paper ( Held On : 2 Dec 2016 )

Answer 1

Given

\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3x^2,}&o\le x\le1\\ {0,}&{elsewhere} \end{array}} \right.\)

Calculation

P(X < a) = 1/a² = \(\mathop \smallint \nolimits_0^a f\left( x \right)dx = 1/2\)

⇒ \(3\mathop \smallint \nolimits_0^a\)x²dx = 1/2

⇒ 3[x³/3]^a₀ = 1/2

⇒ 3 × (a³/3 - 0) = 1/2

⇒ a³ = 1/2

∴ The value of a such that P[x ≤ a] = P[x > a] is (1/2)^1/3