Q. 7.52 Show how to compute Cov(X,Y) fro... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q. 7.52 (page 363)

Show how to compute $Cov (X, Y)$ from the joint moment generating function of $X$ and $Y$ .

Short Answer

Expert verified

The Compute $Cov (X, Y)$ from the joint moment generating function value are $Cov (X, Y) = (\frac{{鈭�}^{2}}{鈭� t_{1} 鈭� t_{2}} M_{X, Y} - \frac{鈭�}{鈭� t_{1}} M_{X, Y} \frac{鈭�}{鈭� t_{2}} M_{X, Y}) (0, 0)$ .

Step by step solution

Given Information

The joint moment generating function of $X$ and $Y$ .

Given Information

We have that the joint moment generating function of $X$ and $Y$ is,

$M_{X, Y} (t_{1}, t_{2}) = E (e^{t_{1} X + t_{2} Y})$

Observe that,

$\frac{鈭�}{鈭� t_{1}} M_{X, Y} (t_{1}, t_{2}) = E (X e^{t_{1} X + t_{2} Y})$

Which implies

$E (X) = \frac{鈭�}{鈭� t_{1}} M_{X, Y} (0, 0)$ and

$E (Y) = \frac{鈭�}{鈭� t_{2}} M_{X, Y} (0, 0)$ .

Explanation

Now, consider what happens if we partially differentiate $M_{X, Y} (t_{1}, t_{2})$ respective to $t_{1}$ and then to $t_{2}$ . We end up with

$\frac{{鈭�}^{2}}{鈭� t_{1} 鈭� t_{2}} M_{X, Y} (t_{1}, t_{2}) = E (X Y e^{t_{1} X + t_{2} Y})$

Which implies,

$\frac{{鈭�}^{2}}{鈭� t_{1} 鈭� t_{2}} M_{X, Y} (0, 0) = E (X Y)$

Finally we have that, $Cov (X, Y) = E (X Y) - E (X) E (Y)$

$= (\frac{{鈭�}^{2}}{鈭� t_{1} 鈭� t_{2}} M_{X, Y} - \frac{鈭�}{鈭� t_{1}} M_{X, Y} \frac{鈭�}{鈭� t_{2}} M_{X, Y}) (0, 0)$ .

Final answer

The $Cov (X, Y)$ joint moment generating function value are $Cov (X, Y) = (\frac{{鈭�}^{2}}{鈭� t_{1} 鈭� t_{2}} M_{X, Y} - \frac{鈭�}{鈭� t_{1}} M_{X, Y} \frac{鈭�}{鈭� t_{2}} M_{X, Y}) (0, 0)$ .

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91影视

Show how to compute $Cov (X, Y)$ from the joint moment generating function of $X$ and $Y$ .

Short Answer

Step by step solution

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Explanation

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91影视

Show how to compute Cov(X,Y) from the joint moment generating function ofX and Y.

Short Answer

Step by step solution

Given Information

Given Information

Explanation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Discrete Mathematics

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Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Show how to compute $Cov (X, Y)$ from the joint moment generating function of $X$ and $Y$ .