Final answer:
To find the probability that 0 < x < 2 using the given joint density function, we integrate over the range 0 to 2 for x and 0 to ∞ for y.
Step-by-step explanation:
The given joint density function is fX,Y(x,y) = 2e-(x+y) for x > 0, y > 0.
To find the probability that 0 < x < 2, we need to integrate the joint density function over the range 0 to 2 for x and 0 to ∞ for y.
P(0 < x < 2) = ∫02 ∫0∞ 2e-(x+y) dy dx
By integrating, we get P(0 < x < 2) = 1 - e-2 ≈ 0.865.