Chapter 7: Q.7.77 (page 358)

The joint density of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$ ,
$- \infty < x < \infty$
(a) Compute the joint moment generating function of $X$ and $Y$ .
(b) Compute the individual moment generating functions.

Short Answer

Expert verified

a). The joint moment generating function of $X$ and $Y$ are $M_{(X, Y)} (t_{1}, t_{2}) = \exp (\frac{t_{1}^{2}}{2}) \frac{1}{(1 - t_{2} - t_{1})}$ .

b). The individual moment generating functions are $M_{X} (t_{1}) = \exp (\frac{t_{1}^{2}}{2}) {(1 - t_{1})}^{- 1}$ and $M_{Y} (t_{2}) = {(1 - t_{2})}^{- 1}$ .

Step by step solution

Given Information (Part a)

$\begin{array}{r} f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty \\ - \infty < x < \infty \end{array}$

Explanation (Part a)

$M_{(X, Y)} (t_{1}, t_{2}) = E [\exp (t_{1} x + t_{2} y)]$

$= \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} \int_{- \infty}^{\infty} \exp (t_{1} x + t_{2} y) \exp (- y) \exp (\frac{- (x - y)^{2}}{2}) d x d y$

$= \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} \int_{- \infty}^{\infty} \exp (t_{1} x + t_{2} y - y - \frac{x^{2}}{2} - \frac{y^{2}}{2} + x y) d x d y$

$= \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} \exp (t_{2} y - y - \frac{y^{2}}{2}) (\int_{- \infty}^{\infty} \exp (\frac{- 1 (x^{2} - 2 b x)}{2}) d x) d y$

$= \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} \exp (t_{2} y - y - \frac{y^{2}}{2}) (\int_{- \infty}^{\infty} \exp (\frac{- (x - b)^{2}}{2}) \exp (\frac{b^{2}}{2}) d x) d y$

Explanation (Part a)

$= \frac{1}{\sqrt{2 π}} \int_{0}^{\infty} \exp ((t_{2} + t_{1} - 1) y + \frac{t_{1}^{2}}{2}) \sqrt{2 π} d y$

$= \int_{0}^{\infty} \exp (\frac{t_{1}^{2}}{2}) \exp (- y (1 - t_{2} - t_{1})) d y$

$= \exp (\frac{t_{1}^{2}}{2}) \int_{0}^{\infty} \exp (- y (1 - t_{2} - t_{1})) d y$

$= {\exp (\frac{t_{1}^{2}}{2}) \frac{\exp (- y (1 - t_{2} - t_{1}))}{(- (1 - t_{2} - t_{1})}|}_{0}^{\infty}$

Therefore,

$M_{(X, Y)} (t_{1}, t_{2}) = \exp (\frac{t_{1}^{2}}{2}) \frac{1}{(1 - t_{2} - t_{1})}$

Final Answer (Part a)

The joint moment generating functions of $X$ and $Y$ are

M_{(X, Y)} (t_{1}, t_{2}) = \exp (\frac{t_{1}^{2}}{2}) \frac{1}{(1 - t_{2} - t_{1})}

Given Information (Part b)

$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$

$- \infty < x < \infty$

Explanation (Part b)

If $t_{2} = 0$ we have

$M_{x} (t_{1}) = \exp (\frac{t_{1}^{2}}{2}) \frac{1}{(1 - 0 - t_{1})}$

$= \exp (\frac{t_{1}^{2}}{2}) {(1 - t_{1})}^{- 1}$

If $t_{1} = 0$ we have

$M_{Y} (t_{2}) = \exp (\frac{0}{2}) \frac{1}{(1 - t_{2} - 0)}$

$= {(1 - t_{2})}^{- 1}$

Final Answer (Part b)

Individual moment generating functions,

If $t_{2} = 0$ , $M_{x} (t_{1}) = \exp (\frac{t_{1}^{2}}{2}) \frac{1}{(1 - 0 - t_{1})}$

If $t_{1} = 0$ , $M_{Y} (t_{2}) = {(1 - t_{2})}^{- 1}$ .

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The joint density of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$ ,
$- \infty < x < \infty$
(a) Compute the joint moment generating function of $X$ and $Y$ .
(b) Compute the individual moment generating functions.

Short Answer

Step by step solution

Given Information (Part a)

Explanation (Part a)

Explanation (Part a)

Final Answer (Part a)

Given Information (Part b)

Explanation (Part b)

Final Answer (Part b)

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The joint density of X and Y is given byf(x,y)=12πe-ye-(x-y)2/20<y<∞,-∞<x<∞(a) Compute the joint moment generating function of X and Y.(b) Compute the individual moment generating functions.

Short Answer

Step by step solution

Given Information (Part a)

Explanation (Part a)

Explanation (Part a)

Final Answer (Part a)

Given Information (Part b)

Explanation (Part b)

Final Answer (Part b)

One App. One Place for Learning.

Most popular questions from this chapter

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

The joint density of $X$ and $Y$ is given by
$f (x, y) = \frac{1}{\sqrt{2 π}} e^{- y} e^{- (x - y)^{2} / 2} 0 < y < \infty$ ,
$- \infty < x < \infty$
(a) Compute the joint moment generating function of $X$ and $Y$ .
(b) Compute the individual moment generating functions.