922 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 



here 



G = H P(s, r) log 



a(s/r) 



- J2v(s, r) log a(s/r) + H{X) 



rs 



where H{X) is the source rate as defined by Shannon. The first term is 

 maximized by putting 



^kP{k, r) q{r) 



Then (7max = H(X) — H{X/Y), which is the rate of transmission de- 

 fined by Shannon. 



WHEN THE ODDS ARE NOT FAIR 



Consider the case where there is no track take, i.e., 



but where as is not necessarily 



1 



V{s) 



It is still permissible to set ^s a{s/r) = 1 since the gambler can effec- 

 tively hold back any amount of money by betting it in proportion to 

 the I /as . Equation (1) now can be written 



G = ^ P(s, r) log a(s/r) -f- J2 Pi^) log a. . 



rs s 



G is still maximized by placing a(s/r) = q{s/r) and 



G^max = -H{X/Y) + Y. Pis) log as 



s 



= H(a) - HiX/Y) 









where 



H{a) = X pis) log as 



Several interesting facts emerge here 



(a) In this case G is maximized as before by putting a{s/r) ^ qis/r). 

 That is, the gambler ignores the posted odds in placing his bets! 



