GATE 2015 EE – Question 11
A random variable $X$ has probability density function $f(x)$ as given below:
$$f(x) = \begin{cases} a + bx & \text{for } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$$
If the expected value $E[X] = 2/3$, then $Pr[X < 0.5]$ is ________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.24 to 0.26
Explanation
The density must integrate to 1, so $a + \frac{b}{2} = 1$. The mean is $\int_0^1 x(a + bx)\,dx = \frac{a}{2} + \frac{b}{3} = \frac{2}{3}$. Substituting $a = 1 - \frac{b}{2}$ gives $\frac{1}{2} + \frac{b}{12} = \frac{2}{3}$, so $b = 2$ and $a = 0$. Then $Pr[X < 0.5] = \int_0^{0.5} 2x\,dx = 0.25$.