Given that,

X and Y are two random variables

The joint pdf of them is given by,

\(f(x,y)=\begin{cases}x+y\ \ \ 0<x<1, 0<y<1\\0 \ \ \ \ \ otherwise\end{cases}\)

By definition of cumulative distribution function,

\(F(x,y)=\int_{0}^{x}\int_{0}^{y}(s+t)dtds\)

\(=\int_{0}^{x}s(t)_{0}^{y}+(\frac{t^{2}}{2})_{0}^{y}ds\)

\(=\int_{0}^{x}sy+\frac{y^{2}}{2}ds\)

\(=y(\frac{s^{2}}{2})_{0}^{x}+\frac{y^{2}}{2}(s)_{0}^{x}\)

\(=\frac{yx^{2}}{2}+\frac{xy^{2}}{2}\)

Now, \(F(\frac{1}{4}, \frac{1}{2})=\frac{(\frac{1}{2})(\frac{1}{4})^{2}}{2}+\frac{(\frac{1}{2})^{2}(\frac{1}{4})}{2}=\frac{3}{64}\)