If a continuous random variable x has the probability density function

$f (x) = {\begin{matrix} 3 x^{2}, & o \leq x \leq 1 \\ 0, & e l s e w h e r e \end{matrix}$

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

$f (x) = {\begin{matrix} 3 x^{2}, & o \leq x \leq 1 \\ 0, & e l s e w h e r e \end{matrix}$

Calculation

P(X < a) = 1/a² = $\int_{0}^{a} f (x) d x = 1 / 2$

⇒ $3 \int_{0}^{a}$ 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