IGNOU MTE 11 2026 - Consider the joint probability density function$Are both x a

Question

Consider the joint probability density function
$$f(x, y) = y^2 e^{-y(x+1)} ; x \ge 0, y \ge 0.$ $
Are both x and y regressions linear? Give reasons for your answer.

09 Jan 2026

Answer :

Word Count : 311

The given joint probability density function is [ f(x, y) = y^2 e^{-y(x+1)}, \quad x \ge 0, \ y \ge 0. ] To check if the regressions of (Y) on (X) and (X) on (Y) are linear, we need (E(Y|X=x)) and (E(X|Y=y)). Step 1: Marginal of (X) [ f_X(x) = \int_0^\infty y^2 e^{-y(x+1)} , dy ] Use the formula (\int_0^\infty y^2 e^{-ay} dy = \frac{2}{a^3}), with (a =

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