Question

A continuous random variable X has a probability density function \[ f(x) = \begin{cases} 3x^{2}, & \text{if } 0 \leq x \leq 1 0, & \text{otherwise} \end{cases} \] If \(P(X \leq a) = P(X > a)\), then the value of a is equal to _______

• A. \(1/33^{1/3}\)
• B. \(1/32^{1/3}\)
• C. \(1/22^{1/3}\)
• D. \(1/2^{1/3}\)

Hint:

The point $a$ where $P(X \leq a) = 0.5$ is by definition the median of the distribution.
For a power function distribution $f(x) = k x^{k-1}$ on $[0,1]$, the cumulative distribution function is $F(x) = x^k$.
Setting $F(a) = 0.5$ gives $a^k = 0.5 \implies a = (1/2)^{1/k}$ directly.

Answer 1

The Correct Option is D