Let us derive more explicitly an example of point masses in the limit of nth order statistics for uniform variables. Consider the i.i.d. distribution (X1, . . . ,Xn) where each X; N U [0, 1]. It is simple to see, generalizing from the case where n = 5 that Prof. Ellison computed, that the pdf of the nth order statistic Y = max (X1 , . . . , X\") is fy (y) = ny"'1. Now we are interested in the behavior of fl, (3;) in the limit to ) oo (read: \"n goes to positive innity\"). We can nd this by considering two regions in the support ofy (which is [0, 1]}. One is the region [0, 1) (Le. excluding y = 1 }, and second is the single point 'region' 3; = 1. For the rst region, for any fixed 1; E [0, 1), y\"'1 goes to 0 as n > 00. More over it actually tends to 0 "faster\" than n tends to 00 (this can be shown by L'Hospital's rule, but is outside the scope of this course!). Therefore for y E [0, 1), f9. = n a: gun1 l- 0 in the limit in ) oo; in other words, for this region the pdf tends to 0 for this region! Question 2 0.0\" .0 point (graded) Please ll in the blank for y = 1. For the second. single point "region" 11; = 1, where does I}, = n t y\"_1 tend to? Simplify and express your answer in terms of n. Submit You have used 0 of 2 attempts Save Just for fun: Draw out fy (y) for all y E [0, 1] in the limit In > 00. Do you see why we say these are "point masses in the limit" of nth order statistics? Discuss below