IGNOU BMTC 101 2026 - . Consider the joint probability density function

Question

. Consider the joint probability density function

$$f(x, y) = y^2 e^{-y(x+1)}; x \geq 0, y \geq 0.$

Are both x and y regressions linear? Give reasons for your answer.

09 Jan 2026

Answer :

Word Count : 267

To check whether the regressions are linear, we first need the conditional expectations (E[Y \mid X=x]) and (E[X \mid Y=y]). The joint pdf is: [ f(x, y) = y^2 e^{-y(x+1)}, \quad x \ge 0, y \ge 0. ] Step 1: Find the marginal pdfs. (a) Marginal of X) [ f_X(x) = \int_0^\infty f(x,y), dy = \int_0^\infty y^2 e^{-y(x+1)} , dy. ] Use the formula (\int_0^\infty y^2 e^{-a y} dy = \frac{2}{a^3}): [ f_X(x) = \frac{2}{(x+1)^3}, \quad x \ge 0. ] (b) Marginal of Y) [ f_Y(y) = \int_0^\infty

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