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We are given the joint probability density function (PDF): \[ f(x,y) = y^2 e^{-y(x+1)}, \quad x \geq 0, y \geq 0. \] To analyze whether both \( x \) and \( y \) regressions are linear, we need to compute the conditional expectations: \[ E[Y | X = x] \quad \text{and} \quad E[X | Y = y] \] and check if they are linear functions of \( x \) and \( y \), respectively. ### Step 1: Find the Marginal PDFs #### Marginal PDF of \( X \) \[ f_X(x) = \int_0^\infty f(x,y) \, dy = \int_0^\infty y^2 ________ _____ _______ _______ ______ __________ _________ _________ _____.
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