AMS 311 - Stony Brook
AMS 311, Lecture 20
April 17, 2001
Examination 2 will be given Thursday, April 19. It will cover Chapters 1 through 8. You may use one text; you may write in that text; and you can have papers stapled or taped into the text. You may use a calculator. You may not share notes or calculators with other students. There will be eight or so problems. Remember that there were four problem types in the last two quizzes: defining and calculating the expectation of a function; univariate transformations of a random variable (really a calculus problem in change of variables in integration); finding a conditional pdf; and bivariate transformations of random variables (also a calculus problem in change of variables in multivariate integration). The examination is also comprehensive, and so I will probe the more difficult problems in the last examination. One is traditionally Bayes’ theorem (as a class you did well on this in the last examination). Another area is that of “stopping rule” problems; this did cause difficulty. There is also the problem of finding cdfs, pdfs, means and variances of random variables (see last problem of the first exam). These problems apply to conditional distributions. Finally, remember that the gamma integral is a powerful tool for finding moments of exponential and gamma distributions.
Example 1
The joint probability function (pdf) of the random variables (X,Y) is [pic] and zero otherwise. Find the marginal probability function (pdf) of Y. Be sure to specify the range on Y. Find the conditional probability function (pdf) of X|Y=y. Be sure to specify the range.
Additional examples:
Under the specification above, what is [pic] What is [pic]
Example 2
The joint probability function (pdf) of the random variables (X,Y) is [pic] and zero otherwise. That is, the random variables X and Y are independent exponential random variables, each with parameter (. Find the joint probability function (pdf) of the random variables W=X and Z=X/Y. Be sure to include the range of the random variables in your answer.
Additional example problems are:
Using the specification in example 2, find the marginal distribution of Z. This is problem 14 on page 313.
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