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l l I T A ~ - - ~ | ' [ = - ] i l l ~ - i | [ i ' ] ~ - l l l ~ | : - ] i ( ' = l i i l i i [ = - ] ' i 4 " - | M i c h a e l K l e b e r a n d R a v i V a k i l , E d i t o r s
Expected Value Road Trip SAM VANDERVELDE
This c o l u m n is a p l a c e f o r those b i ts
o f c o n t a g i o u s m a t h e m a t i c s t h a t t rave l
f r o m p e r s o n to p e r s o n in the
c o m m u n # y , b e c a u s e they are so
e legant , supr i s ing , o r a p p e a l i n g t h a t
o n e h a s a n urge to p a s s t h e m on.
C o n t r i b u t i o n s a r e m o s t we lcome .
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s a result of a questionable de- cision to combine a cross-coun-
~ ~*try move with a four-week stay at a summer math program, I found my- self driving solo from San Francisco to Salt Lake City in a single day during the summer of 2007. To combat the lethargy of the midafternoon stretch, I decided to reconstruct mentally the solution to a classic problem that dates back at least to Laplace, according to Uspensky [ 1 ] - namely, to determine on average how many numbers, chosen uniformly at random from the interval [0, 1], one must select before their sum exceeds 1. The delightful answer, as we all know, is e.
This question has been the subject of renewed interest lately. In [2], Curgus and Jewett examine the function o~(t), defined as the expected number of random se- lections from [0, 1] that must be made before the cumulative total exceeds t. They go on to find an explicit formula for oe(t) using the theory of delay func- tions. It stands to reason that a(t) ~ 2t,
since each selection is equal to 1, on av- erage. Their surprising observation is that
2 lim(a(t) - 2t) = -- . t+~ 3
One can independent ly show that the value of oL(t) may be expressed in terms of powers of e for integral values of t. We are thus led to statements such as
e6 _ 5e 5 + 8e i _ 9e 3 + 2e 2 e 2 3 120
12},
by considering c~(6), for example. The two quantities agree to six digits past the decimal point�9
Our purpose here is to pursue a thought which struck me as yet another low mountain range passed by to the south somewhere in Nevada. We wish to know how many numbers, chosen uni- fon-nly at random from the interval [1, d, one must select before their product ex- ceeds e. To cut the suspense, the answer is the somewhat unlikely expression
1 - - e - 1 e e-1 + -
e Readers may test their intuition by pre- dicting whether this quantity is greater than or less than e.
The assault proceeds in predictable fashion: For n--> 1, let q,2 be the prob- ability that a product of n numbers cho- sen from [1, e] is no t greater than e. (It will be convenient to define q0 = 1 as well.) Then the probability that the product exceeds e for the first time at the nth selection is
( 1 - q . ) - ( 1 - q . - 1 ) = q n - 1 - q . ,
valid for n -> 1. The expected value we seek is
n( qn_ 1 - qn) n = 1
= Zq,,-1 + ~ ( n - - l)qn-i -- ~ n q n n = l n = l n : l
= ~ q-. t / = O
The fact that all sums converge ab- solutely will be clear once we obtain a closed form expression for qn. The tick- lish part is finding the formula.
The region R,, in the n-cube [1, d n consisting of points (& . . . . , xu), the product of whose coordinates is at most e, is described by
l <--Xl <--e
e 1 < x 2 _ - -
xl
i
e
1 ~ X n < -- X l . . �9 X n - 1
We may then compute q , via
_ 1 fR qn (e - 1)'l dxn �9 �9 �9 dXl.
We focus on the integral by defining
O, = f R,, dxn" �9 �9 d&. The first six val- u e s a r e e - 1, 1, L e - 1 - k e + 1, k e - -
2 ' 3 8 1, and - U e + 1, found by electronic .~0 means. Evaluating these integrals di-
�9 2008 Springer Science+Business Media, Inc., Volume 30, Number 2, 2008 1 7
rectly becomes an increasingly arduous affair as n grows, so it comes as a pleas- ant surprise to discover that their value may be ascer ta ined with hardly any ef- fort. Indeed, we find that
= _e_ 1) dxn ' ' ' do:], On+ l fR. ( Xl " x.
by performing the easy integration with respect to x,+~. It follows that
O n + l qL On : f r e d ~ n . . . d ~ l . Jle n X l " " " Xn
Making the change of variables Yk = = • dx~, we find that In xk so that dyk xk
+ On= f e d y l ' ' ' 0,,+ clyn. 1 ate h
It is clear that the region R" consists of those points (Jl . . . . . Yn) satisfying Yk -> 0 and .)21 q- �9 �9 �9 "b Yn ~ 1, that is, R;, is a right unit n-simplex. It is well- known that the volume of this region is • so we have shown that n! 1
e 0n+l + On n!
It is conceivable that one might s tumble upon this approach while nav- igating northern Nevada by car. In fact, I initially headed down another road by asking how many numbers from the interval [1, 2] wou ld be requi red to ob- tain a product that e x c e e d e d 2. Several mental integrations later it became clear that I had taken a wrong turn when the accumulat ion of ln2's became over- whelming. By the time the quest ion had been proper ly formulated, the Great Salt Flats beckoned , and the in- teresting task of answering the ques- tion was p o s t p o n e d until a later time.
Resuming the argument, we now find an expression for fl,,. For n --> 1 let bn be the nth partial sum of the usual series for e - I , so that
b n = l - l + - - 1 . . . . 1 2
1 + ( - 1 ) n-] - - ( n - 1)!"
We claim that 0 , ,= ( - 1 ) n ( 1 - b,e). The quantit ies agree for n = 1, and for n -> 2 we compute
0n + O,,+] = ( -1 )n (1 - bne) + ( - 1 ) ' + 1 ( 1 - bn+le)
= e(-1)n(bn+l - bn)
= e(--l)n{(--1)n I t " ! /
e n!'
as desired. This provides the sought- after express ion for q,, namely
( - 1)n(1 - b,,e) qn = ( e - 1) n
for n - 1 . Clearly [ q n [ - ( e - 1 ) -~', which justifies the manipulat ions of the infinite series above.
The final s tep of summing the q,, presents a very satisfying exercise in- volving geometr ic and exponent ia l se- ries. To begin,
2 2 q~, = 1 + ( - 1)" ,,=0 ,= l ( e - 1) n
- Z (-1)n(bne) , = ] ( e - 1) u
v~ 1 ~ ( -1)n(bn e)
e n=l/' ( e - 1) n
We evaluate the remaining term by writing b, as a sum and interchanging the order of summations, obtaining
_ ~ (-1)n(bne) ,,=, i-i7
( - 1 ) k = - e ~ .
n=] k=O
2 ? = --e k! (e Z - ~ n k=O n = k + l
~__0 (--1) k (--1) k+] = - -e = k! e ( e - 1) k
= s 1 1
k=o k! ( e - 1) k 1
~ e e - 1
This completes the computat ion. It is only natural to speculate
whether the phe nome non descr ibed by Curgus and Jewett occurs in this con- text. So let flU) be the expec ted num- ber of random selections that must be made from the interval [1, d before the product exceeds e t. Since the average value of In x on the interval [1, el is e-11 we anticipate that
t /3(1) ~ e e-1
But can a more precise s tatement be made regarding the relat ionship be-
tween these two quantities? In the spirit of the result ment ioned previously, one might hope for a limiting ratio, such as
t
lira /3(0" e e-1 = C, t --->0r
for some constant C > 1. However , it is not yet clear that the limit exists, and if so, what an exact value for C might be. Of course, so far we have establ ished that
1 - - e - 1
/3(1) - e ~'-1 - e
which now seems slightly mysterious, given that a difference appears , rather than a ratio.
Incidentally, since lnx is concave down, a number x chosen uniformly from [1, el will yield a value for lnx that is closer to 1, on average, than we would have obta ined by simply having chosen a number uniformly from [0, 1]. In other words, a random selection from [1, el contr ibutes more to a product than a selection from [0, 1] will contribute to a sum. So we expect that fewer selec- tions are required for our product to ex- ceed a given value, implying that/3(1) < a(t) for all posit ive t. As anticipated,
/3(1) ~ 2.42169 < e,
settling the matter raised earlier. And with this observation, we conclude our mathematical journey, or at least this leg of the trip.
ACKNOWLEDGMENTS
I am grateful to Michael Sheard for read- ing through this piece and making sev- eral helpful suggestions. I also wish to thank the editor for encouraging me to document this mathematical morsel.
REFERENCES
[1] J. Uspensky, Introduction to Mathematical Probability, New York, NY, 1937.
[2] B. Curgus and R. I. Jewett, An unexpected
limit of expected values, Expo. Math. 25
(2007), 1-20.
Sam Vandervelde Department of Mathematics, CS & Statistics St. Lawrence University 23 Ramada Drive Canton, NY 13617 e-mail: [email protected]
18 THE MATHEMATICAL INTELLIGENCER
