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

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,

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

Which implies

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

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

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

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

Which implies,

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

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

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

Final answer

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

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

Short Answer

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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

Geometry

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Study anywhere. Anytime. Across all devices.

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