By Marcus R.

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**Extra resources for A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution**

**Sample text**

We call i the toss index, which takes values 1, . . , n. A typical sequence of n tosses would be H1 ∩ H2c ∩ H3 ∩ · · · ∩ Hn−1 ∩ Hnc , where Hic is the event that the ith toss is tails. The probability that n tosses result in k heads and n − k tails is P H1 ∩ · · · ∩ Hn , where Hi is either Hi or Hic , and the union is over all such intersections for which Hi = Hi occurs k times and Hi = Hic occurs n − k times. Since this is a disjoint union, P H1 ∩ · · · ∩ Hn = ∑P H1 ∩ · · · ∩ Hn . By independence, P H1 ∩ · · · ∩ Hn = P H1 · · · P Hn = pk (1 − p)n−k is the same for every term in the sum.

The ﬁrst group of zeros tells how many apples, the second group of zeros tells how many bananas, and the third group of zeros tells how many carrots. The ones are used to separate the groups of zeros. As another example, (0, 0, 0, 1, 0, 1, 0) means three apples, one banana, and one carrot. How many customer choices are there? Solution. The question is equivalent to asking how many 7-bit sequences there are with 7 = 75 = 72 . ﬁve zeros and two ones. 47 (unordered sampling with replacement). Suppose k numbers are drawn with replacement from the set A = {1, 2, .

Km−1 ! the multinomial coefﬁcient. When m = 2, n k0 , k1 = n k0 , n − k0 = n! = k0 ! (n − k0 )! n k0 becomes the binomial coefﬁcient. Unordered sampling with replacement Before stating the problem, we begin with a simple example to illustrate the concepts involved. 46. An automated snack machine dispenses apples, bananas, and carrots. For a ﬁxed price, the customer gets ﬁve items from among the three possible choices. For example, a customer could choose one apple, two bananas, and two carrots.

### A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution by Marcus R.

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