Question

The joint probability density function of random variables X and Y is given as follows: f(x, y) = 8ye^{-2x} \quad y < x < \infty, y > 0. a) Show that the joint moment generating function (mgf) of X and Y is \frac{8}{(2 - t_1)(2 - t_1 - t_2)^2} \quad \text{where } t_1 < 2, \quad t_1 + t_2 < 2 b) Hence, use the joint moment generating function (mgf) to find i) E[X] and E[Y]. Are X and Y independent? ii) E[XY]. iii) Var(X - 2Y).

          The joint probability density function of random variables X and Y is given as follows:
f(x, y) = 8ye^{-2x} \quad y < x < \infty, y > 0.
a) Show that the joint moment generating function (mgf) of X and Y is
\frac{8}{(2 - t_1)(2 - t_1 - t_2)^2} \quad \text{where } t_1 < 2, \quad t_1 + t_2 < 2
b) Hence, use the joint moment generating function (mgf) to find
i) E[X] and E[Y]. Are X and Y independent?
ii) E[XY].
iii) Var(X - 2Y).

The joint probability density function of random variables X and Y is given as follows:
f(x, y) = 8ye^-2x y < x < ∞, y > 0.
a) Show that the joint moment generating function (mgf) of X and Y is
(8)/((2 - t1)(2 - t1 - t2)^2) where t1 < 2, t1 + t2 < 2
b) Hence, use the joint moment generating function (mgf) to find
i) E[X] and E[Y]. Are X and Y independent?
ii) E[XY].
iii) Var(X - 2Y).

Added by Rebecca M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Ekaveera Kumar

01/28/2022

Step 1

First, we integrate with respect to x: $$M(t_1, t_2) = \int_{0}^{\infty} 8ye^{t_2y} \left[\frac{e^{t_1x}}{t_1}\right]_{x=y}^{x=0} dy$$ $$M(t_1, t_2) = \int_{0}^{\infty} 8ye^{t_2y} \left(\frac{1}{t_1} - \frac{e^{t_1y}}{t_1}\right) dy$$ Now, we integrate with Show more…

Show all steps

Thanks for your feedback!

The joint probability density function of random variables X and Y is given as follows: f(x,y) = 8ye^(-x) 0 < x < Y, Y > 0, Show that the joint moment generating function (mgf) of X and Y is M(t1,t2) where t1 < 2, t1 + t2 < 2 (2 - t1)(2 - t2)^2 / 6) Hence, use the joint moment generating function (mgf) to find E[X] and E[Y]. Are X and Y independent? E[XY]. Var(X + 2Y).