GATE 2018 | MATHS | Q-33

Question

Let \(X\) and \(Y\) have a joint probability density function given by
\[ f_{X,Y}(x, y) = \begin{cases}
2 & \text{if } 0 \leq x \leq 1 - y \text{ and } 0 \leq y \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]

If \(f_Y\) denotes the marginal probability density function of \(Y\), then \(f_Y(1/2)\) is given by

Answer 1

If \( f_Y \) denotes the marginal probability density function of \( Y \), then \( f_Y(1/2) \) is given by
\[ f_Y(1/2) = \int_{-\infty}^{\infty} f_{X,Y}(x, 1/2) \, dx \]
\[ = \int_{0}^{1/2} 2 \, dx \]
\[ = [2x]_{0}^{1/2} \]
\[ = 1 \]