Stat 5101 Lecture Notes - School of Statistics

20 Stat 5101 (Geyer) Course Notes• cannot be a function of the dummy variable of integration x, and• is a function of the free variable y.Thus∫f X,Y (x, y) dx = some function of y onlyand hence can only be f Y (y) and cannot be f X (x). Thus making the mistakeof integrating with respect to the wrong variable (or variables) in attemptingto produce a marginal is really dumb on two counts: first, you were warnedbut didn’t get it, and, second, it’s not only a mistake in probability theory butalso a calculus mistake. I do know there are other reasons people can makethis mistake, being rushed, failure to read the question, or whatever. I knowsomeone will make this mistake, and I apologize in advance for insulting youby calling this a “dumb mistake” if that someone turns out to be you. I’m onlytrying to give this lecture now, when it may do some good, rather than later,written in red ink all over someone’s test paper. (I will, of course, be shockedbut very happy if no one makes the mistake on the tests.)Of course, we sum out discrete variables and integrate out continuous ones.So how do we go from f W,X,Y,Z to f X,Z ? We integrate out the variables wedon’t want. We are getting rid of W and Y ,so∫∫f X,Z (x, z) = f W,X,Y,Z (w, x, y, z) dw dy.If the variables are discrete, the integrals are replaced by sumsf X,Z (x, z) = ∑ ∑f W,X,Y,Z (w, x, y, z).w yIn principle, it couldn’t be easier. In practice, it may be easy or tricky, dependingon how tricky the problem is. Generally, it is easy if there are no worries aboutdomains of integration (and tricky if there are such worries).Example 1.5.1.Consider the distribution of Example 1.3.1 with joint density of X and Y givenby (1.22). What is the marginal distribution of Y ? We find it by integratingout X∫∫ 1∣f Y (y) = f(x, y) dx = (x + y) dx = x2 ∣∣∣1 ( ) 12 + xy =2 + y0Couldn’t be simpler, so long as you don’t get confused about which variableyou integrate out.That having been said, it is with some misgivings that I even mention thefollowing examples. If you are having trouble with joint and marginal distributions,don’t look at them yet! They are tricky examples that very rarely arise.If you never understand the following examples, you haven’t missed much. Ifyou never understand the preceding example, you are in big trouble.0