Let the random variable M n denote the number of males in the cons- anguinity after n weeks. The ratio. We next review briefly the fundamentals of conditional expectation and martingales, a simplification of sections 8. We cite the three laws of conditional expectation, a simplified version of Proposition 8. Imagine that you are playing roulette in a fair casino where there is no zero on the wheel. For a consanguinity, the number of male members M n , which can never decrease, is not a martingale.
With probability 1 4 , a new male member is selected, and the proportion of males rises to 2 5. With probability 3 4 , a new female member is selected, and the proportion of males drops to 1 5. The general proof is scarcely more difficult. Introduce a new random variable X n , the number of male members chosen in week n. This analysis appears in almost every introduction to martingales; for example, as exercise 27 on page of . What appears not to be well known is that there is also a second-degree martingale that makes the computation of variance equally easy.
The inspiration for this second martingale is a symmetric random walk, as described, for example, in Example 9. Again condition on Y n. It took a bit of experimentation for the author to discover a second degree martingale even for the simple special case that appeared on the examination. Again we condition everything on M n. The key is that. The Martingale Convergence Theorem, stated and proved, for example, in section 9.
From the results that we have just. Any book on mathematical statistics, e. The expecta- tion is. So the conjectured beta distribution has the right expectation and variance. Rather than attempt to compare higher moments, though, it will simpler just to work out the probability mass function and take its limit.
We have already identified a finite sample space whose elements all have equal probability. The number of elements in the sample space is the product of the number of equally likely choices that can be made in successive weeks: namely. For the first male added there is only one candidate for the member who did the choosing; for the second male added, there are two candidates, and so forth. There are n m ways to select the set of m weeks in which a male is chosen. The probability of adding m males is simply the proportion of elements in the sample space that contribute to this event: namely.
The probability that only females are added over a period of n weeks is simply the ratio. Suppose that the consanguinity agrees to pay a fine of 1 dollar for each week that its membership includes only one male. What is the expected amount of the fine? For example, if the first male is added in week 3, the first two of these indicator functions equal 1, all the others are 0, and the fine is 2 dollars.
But the expectation of 1 n is just the probability of the event that only females are added over a period of n weeks, so. If the consanguinity begins with N members, M males and F females, the proportion of male members is still a martingale:. Furthermore, the second-degree martingale is the simplest imaginable generalization: 4 gets replaced by N.
Again we condition everything on M n and use. With more than one initial male member, the calculation of variance becomes slightly more complicated, since Z 0 depends on the number of initial male members. By the Tower Law 3 ,. I thank the Editor and the referee for their comments. Graduate Studies in Mathematics, 70, p. Oxford University Press, New York. Model in probability theory. For the martingale betting strategy, see martingale betting system.
Main article: Stopping time. Azuma's inequality Brownian motion Doob martingale Doob's martingale convergence theorems Doob's martingale inequality Local martingale Markov chain Markov property Martingale betting system Martingale central limit theorem Martingale difference sequence Martingale representation theorem Semimartingale. Money Management Strategies for Futures Traders. Wiley Finance. Electronic Journal for History of Probability and Statistics. Archived PDF from the original on Retrieved Oxford University Press.
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