Step 1

Given information:

The two continuous random variables X and Y has joint density function defined as:

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

Then,

\(P(X>x, Y>y)=\int_{y}^{1}\int_{x}^{1}f(x,y)dxdy\)

Step 2

\(P(X>0.8, Y>0.8)=\int_{y=0.8}^{1}\int_{x=0.8}^{1}(x+y)dxdy\)

\(=\int_{y=0.8}^{1}\left\{\int_{x=0.8}^{1}(x+y)dx\right\}dy\)

\(=\int_{y=0.8}^{1}\left[\frac{x^{2}}{2}+xy\right]_{0.8}^{1}dy\)

\(=\int_{y=0.8}^{1}\left\{(\frac{1}{2}+y)-(\frac{0.8^{2}}{2}+0.8y)\right\}dy\)

\(=\int_{y=0.8}^{1}\left\{\frac{1}{2}+y-0.32-0.8\right\}dy\)

\(=\int_{y=0.8}^{1}(0.18+0.2y)dy\)

\(=\left[0.18y+\frac{0.2y^{2}}{2}\right]_{0.8}^{1}\)

\(=\left[(0.18+0.1)-(0.18\times 0.8+0.1\times 0.8^{2})\right]\)

\(=0.072\)

The required probability is 0.072.