The Beta PDF. The beta PDF with parameters Q > 0 and {3 >

Chapter , Problem 30

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The Beta PDF. The beta PDF with parameters Q > 0 and {3 > 0 has the form { 1 x o-1 (1 _ X)i3- 1 /x (x) = B(o:, ,8) , 0, if 0:5 x :5 1, otherwise. The normalizing constant is and is known as the Beta function. (a) Show that for any m > 0, the mth moment of X is given by E[X 7n] = B(o: + m, (3) . B(o:, ,8) (b) Assume that 0: and (3 are integer. Show that so that B(o:, (3) = (0: - 1)' ((3 I)! . (0: +(3- 1)! m 0:(0: + 1) . (0: + m - 1) E[X 1 = . ( 0: + (3)(0: + (3 + 1) ... (0: + (3 + m - 1) (Recall here the convention that O! = 1.) Solution. (a) We have E[Xm] = 1 11 m 0-1 (1 _ ) 13- 1 d = B(o + m, )3) B(o, )3) 0 x x x X B(o. )3) 195 (b) In the special case where 0 = 1 or )3 = 1, we can carry out the straightforward integration in the definition of B(o, )3), and verify the result. We will now deal with the general case. Let Y, Yl , ... , Yo+13 be independent random variables, uniformly distributed over the interval [0, 1] , and let A be the event Then, 1 P(A) = (0 +)3+ 1)!' because all ways of ordering these 0 + )3 + 1 random variables are equally likely. Consider the following two events: B = { max{Y1 , ... , Yo } Y}, We have, using the total probability theorem, p(B n C) = /.1 p(BnCI Y = y)Jy (y) dy = /. 1 p( max{Y1 , ... , Yo } y min{Yo+1 , ... , Yo+13 }) dy = /.1 p( max{E . ... , Yo } :-; y) ply :-; min{Yo+J , . .. ' yo+, }) dy = /.' yO(1 _ y)" dy. We also have 1 p(A I BnC) = 0! 1'!' because given the events B and C, all o! possible orderings of Y1 , , Yo are equally likely, and all f'! possible orderings of Yo+! , ... , Yo +13 are equally likely. By writing the equation P(A) = P(B n C) P(A I Bn C) in terms of the preceding relations, we finally obtain ( " + + I ) ' = ", 1 m /.' yO(1 - y)" dy, 1 0< f3 0:. . 1 ' {3' o Y (1 - y) dy = (0: + (3 + I)! . This equation can be written as o:! B! B(o: + 1. {3+ 1) = ( (3 ) " 0:+ + 1 . for all integer 0: > 0, (3 > O.