The continuous random variables (X, Y ) have joint density function fXY (x,y) = xe−xy, x ∈ (0,1), y > 0. a) Derive the marginal density of X and recognise its distribution.

To find the marginal density of X, we need to integrate the joint density function fXY(x, y) over the entire range of y. The marginal density of X, denoted as fX(x), is given by: fX(x) = ∫[0,∞] fXY(x, y) dy Substituting the given joint density function fXY(x, y) = xe^(-xy), we have: fX(x) = ∫[0,∞] xe^(-xy) dy To solve this integral, we can factor out the x term: fX(x) = x ∫[0,∞] e^(-xy) dy Now we can integrate with respect to y: fX(x) = x * [-e^(-xy) / x] evaluated from 0 to ∞ Simplifying further: fX(x) = -e^(-∞) + e^(-0) = 0 + 1 = 1 Therefore, the marginal density of X is fX(x) = 1 for x ∈ (0, 1). Recognizing its distribution, we find that the marginal density of X is a uniform distribution on the interval (0, 1).