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I have a joint density function:

$$f(x,y) \text{ for } x \geq 0, y \geq 0. $$

I'm attempting to find $P\{X + 2Y \geq 1\}$

I have:

$X \geq 1 - 2Y$ $$ \int_1^\infty\int_{1-2y}^\infty f(x,y) \,dxdy. $$ This yields a negative answer which can't be correct. Where should I look from here? I think I'm not setting up the bounds correctly, but I'm not sure how I would find the proper bounds.

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If $y\geq \frac12$ then $1-2y\leq 0$ and you should integrate by $x\in[0,\infty)$.

For $0<y<\frac12$ the inner integral is correct.

Then $$ P(X+2Y>1)=\int_0^\frac12\int_{1-2y}^\infty f(x,y) \,dxdy+\int_\frac12^\infty\int_{0}^\infty f(x,y) \,dxdy. $$

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Perhaps a simpler way of computing $P\{X + 2Y \geq 1\}$ would be to use the complement rule and compute $1-P\{X + 2Y \lt 1\}$, which could be solved by the double integral $$1 -\int_0^1\int_0^{\frac{1}{2} - x/2} f(x,y) \,dy dx $$

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