Question

13. Joint moment generating function. For each of the following joint density functions of the pair (X, Y), find the joint moment generating function $M(s, t) = E(e^{sX+tY})$, and hence find $cov(X, Y)$. (a) We have that $f(x, y) = 2e^{-x-y}$ for $0 < x < y < \infty$. (b) We have that $f(x, y) = \frac{x^2 + y^2 + c^2}{2\pi(2 + c^2)} \exp\{-\frac{1}{2}(x^2 + y^2)\}$, $x, y \in \mathbb{R}$.

          13. Joint moment generating function. For each of the following joint density functions of the pair (X, Y), find the joint moment generating function $M(s, t) = E(e^{sX+tY})$, and hence find $cov(X, Y)$.
(a) We have that $f(x, y) = 2e^{-x-y}$ for $0 < x < y < \infty$.
(b) We have that
$f(x, y) = \frac{x^2 + y^2 + c^2}{2\pi(2 + c^2)} \exp\{-\frac{1}{2}(x^2 + y^2)\}$,
$x, y \in \mathbb{R}$.

$13. Joint moment generating function. For each of the following joint density functions of the pair (X, Y), find the joint moment generating function M(s, t) = E(e^sX+tY), and hence find cov(X, Y). (a) We have that f(x, y) = 2e^-x-y for 0 < x < y < ∞. (b) We have that f(x, y) = (x^2 + y^2 + c^2)/(2π(2 + c^2)){-(1)/(2)(x^2 + y^2)}, x, y ∈ℝ.$

Added by John G.

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