# normal distribution integrating over region.

X and Y are investment returns which are normally distributed with X~N(1000,250) Y~N(1000,250) for two consecutive years. The tax rates for year 1 is 30%. The tax rate for year 2 is 20%. If there are losses the taxes on the losses are returned. What is the probability that the total return is positive after two years.

I am trying to solve as follows:

3 possibilities:

1> x >0, y>0 return positive after two years.

2> x <0, y>0.7x (because 0.3x is returned) return positive after two years.

3> x >0.8y, y<0 (because 0.2y is returned) return positive after two years.

After this I fail to integrate properly for possibility 2 and 3.

Can someone please help me to figure out if there is a nice way to integrate or solve the above problem.

$\mathrm P_\textsf{Answer} =$ $\displaystyle \int_{0}^\infty\int_{0}^\infty f_{X,Y}(x,y)\operatorname d x\operatorname d y \\ \displaystyle +\int_{-\infty}^0\int_{0.7x}^\infty f_{X,Y}(x,y)\operatorname d y\operatorname d x \\ \displaystyle + \int_{-\infty}^0\int_{0.8y}^\infty f_{X,Y}(x,y)\operatorname d x\operatorname d y$
Where $f_{X,Y}(x,y) = \dfrac{\exp\Big(-\frac{(x-1000)^2}{500}-\frac{(y-1000)^2}{500}\Big)}{500 \pi}$
Calculating those last two terms will be a pain, but I put it to you that they are too small to matter, as: $$\int_{0}^\infty\int_{0}^\infty f_{X,Y}(x,y)\operatorname d x\operatorname d y = \tfrac 1 4(\operatorname{erf}(20\sqrt 5)+1)^2 \approx 1$$