# Find the probability P(X+Y<0)?

## In this question i am confused in taking the limit of integration

i did like this for finding the value of constant c i need to do

$$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f_{xy}dxdy=1$$ $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}cxydxdy+\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}2cxydxdy=1$$ now from the question limit value of integration i can change like this $$\int_{-\infty}^{1}\int_{0}^{\infty}cxydxdy+\int_{-1}^{\infty}\int_{-\infty}^{0}2cxydxdy=1$$

but if I will do like I can't get the value of c because integration value will become infinity.

am I taking integration limit wrong?

For the first step: you are 100% correct in trying to set the integral over the domain of $f$ to 1. The limits should be set up as follows: $$\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y) \ dy \ dx = \int_{-1}^0 \int_{-1}^0 2cxy \ dy \ dx + \int_0^1 \int_0^1 cxy \ dy \ dx = 1$$ This can be seen from the domain described by the function given to you. Hopefully you can see how I got the limits I did.
For the second step: try sketching the region on the XY plane that satisfies both $f(x,y) \neq 0$ and $X + Y < 0$. Integrating $f(x,y)$ over this region should give you $P(X+Y<0)$.