Let X and Y be two independent and identically distributed random variables having U(0,1) distribution. Then P(X^2 < Y < X equals _________________ (round off to 2 decimal places)
Since whenever, 0< x < 1 , x^2 < x
\begin{align*}
P (X^2 < Y < X) &= \int_{0}^{1} \int_{x^2}^{x} dy dx \\
&= \int_{0}^{1} (x-x^2) dx \\
&= \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{1} \\
&= \frac{1}{6} = 0.1667
\end{align*}