So, to sum it up swiftly, $X$ symbolizes the likelihood of $X$'s age and if $X$ breaks, $y$ will hopefully live longer. They both have their probability density functions as $e^{-x}$ and $e^{-y}$.
I'm trying to calculate the probability of them exceeding at least the 2-year-mark, but I'm having difficulties.
What I've done thus far is that I've divided the problem into two events:
- X exceeds the 2-year-mark alone, the probability of this happening is $e^{-2}=0.13533...$
- $X$ breaks between 0 and 2 years and $Y$ has to reach at least $2-x$ years of age
For #2 I'm having trouble at deciding what I should use as $f(x)$ when I'm trying to calculate $\iint_{x,y}f(x)dydx$ and I'm not sure if my boundaries are correct.
The boundaries I've brainstormed are as follows:
$0 \lt x \lt 2$
$2-x \lt y \lt \infty$
$x \lt y \lt \infty$
$0 \lt y \lt 2-x$
(The last boundary would be inverse case for $y$ for when $x+y \lt 2$
Can someone help me by telling me how I'm approaching this problem wrong?
"1. X exceeds the 2-year-mark alone, the probability of this happening is $e^{-2}=0.13533\ldots$" i.e.
$$\int_{x=2}^\infty e^{-x} \, dx$$
and similarly
"2. $X$ breaks between $0$ and $2$ years and $Y$ has to reach at least $2-x$ years of age" so
$$\int_{x=0}^2 \int_{y=2-x}^\infty e^{-x}e^{-y}\, dy \, dx$$