Question

The joint distribution function of the random variables X,Y is given by: f(x,y) = xe^{-x(y+1)}?_{(0,?)^2}(x,y). a) Determine the marginal distributions of X and Y. Are X and Y independent? b) Determine the density of X/Y. c) Determine the covariance Cov[X^2 + Y^2 - 2X - 3, Y].

          The joint distribution function of the random variables X,Y is given by:
f(x,y) = xe^{-x(y+1)}?_{(0,?)^2}(x,y).
a) Determine the marginal distributions of X and Y. Are X and Y independent?
b) Determine the density of X/Y.
c) Determine the covariance Cov[X^2 + Y^2 - 2X - 3, Y].

The joint distribution function of the random variables X,Y is given by:
f(x,y) = xe^-x(y+1)?(0,?)^2(x,y).
a) Determine the marginal distributions of X and Y. Are X and Y independent?
b) Determine the density of X/Y.
c) Determine the covariance Cov[X^2 + Y^2 - 2X - 3, Y].

Added by Michael G.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

06/28/2022

Step 1

For X: \[ f_X(x) = \int_{0}^{\infty} xe^{-y-1} dy \] \[ = x \int_{0}^{\infty} e^{-y-1} dy \] \[ = x \left[ -e^{-y-1} \right]_{0}^{\infty} \] \[ = x \left[ 0 - (-e^{-1}) \right] \] \[ = xe^{-1} \] For Y: \[ f_Y(y) = \int_{0}^{\infty} xe^{-y-1} dx \] \[ = Show more…

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The joint distribution function of the random variables X,Y is given by: f(x,y) = xe^{-x(y+1)}χ_{(0,∞)^2}(x,y). a) Determine the marginal distributions of X and Y. Are X and Y independent? b) Determine the density of X/Y. c) Determine the covariance Cov[X^2 + Y^2 - 2X - 3, Y].