Consider two jointly distributed random variables
If the random variables
If the random variables
If the random variables
where the two-dimensional Stieltjes integral may be defined in a manner similar to that in which the one-dimensional Stieltjes integral was defined in section 6 of Chapter 5.
On the other hand,
in the sense that if either of the expectations in (2.5) exists then so does the other, and the two are equal. A rigorous proof of (2.5) is beyond the scope of this book.
In view of (2.5) we have two ways of computing the expectation of a function of jointly distributed random variables. Equation (2.5) generalizes (1.5). Similarly, (1.11) may also be generalized.
Let
The most important property possessed by the operation of expectation of a random variable is its linearity property : if
From (2.5) it follows that
Now
The integral on the right-hand side of
The moments and moment-generating function of jointly distributed random variables are defined by a direct generalization of the definitions given for a single random variable. For any two nonnegative integers
We next define the central moments of the random variables
The covariance derives its importance from the role it plays in the basic formula for the variance of the sum of two random variables :
To prove (2.11), we write
The moments can be read off from the power-series expansion of the moment-generating function, since formally
In particular, the means, variances, and covariance of
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To evaluate the integral in (2.19), let us note that since
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in which
By combining terms in (2.21), we finally obtain that
(2.22)
The covariance is given by
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Thus, if two random variables are jointly normally distributed, their joint probability law is completely determined from a knowledge of their first and second moments, since
Regarded as a random function of time,
Exercises
Exercise 4. 2.1 . An ordered sample of size 5 is drawn without replacement from an urn containing 8 white balls and 4 black balls. For
Answer
Mean,
Exercise 5. 2.2 . An urn contains 12 balls, of which 8 are white and 4 are black. A ball is drawn and its color noted. The ball drawn is then replaced; at the same time 2 balls of the same color as the ball drawn are added to the urn. The process is repeated until 5 balls have been drawn. For
Exercise 6. 2.3 . Let
Answer
Exercise 7. 2.4 . Let Hint : for any real numbers
Exercise 8. 2.5 . Let
Answer
Exercise 9. 2.6 . Let "> in which
Exercise 10. 2.7 . Let "> in which
Answer
Means, 1; variances, 0.5; covariance,
Exercise 11. 2.8 . Let in which
Exercise 12. 2.9 . Do exercise 2.8 under the assumption that
Answer
Means, 4; variances, 6; covariance,
