LONGEST SUCCESS RUNS AND FIBONACCI-TYPE POLYNOMIALS
LONGEST SUCCESS RUNS AND FIBONACCI-TYPE POLYNOMIALS
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最长的成功运行和斐波那契型多项式
DOI：
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发表时间：
1984
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通讯作者：
F. S. Makri
中科院分区：
文献类型：
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作者：
Andreas N. Philippou;F. S. Makri
设Ln是对任意p GB(0，1)具有恒定成功概率p(0 K)(1<；k^n)的n(1)个独立试验的最长成功运行长度。给出了P(Ln^k)‘和P(Ln=k)(0<；k<；n)的计算公式。我们的公式是用多项式系数和k阶斐波那契型多项式给出的(见引理2.1，定义2.1和定理2.1)。作为定理2.1的推论，我们发现了Bollinger[2]的两个计数定理，在他的术语中，涉及长度为n的二进制数的个数，这些二进制数没有(或确实有)一串k个连续的1。我们在第二节中给出了这些结果。在第三节中，我们重新考虑了等待随机变量NK(k^1)，它表示直到k个连续成功发生的Bernoulli试验的次数，并证明了P(Nk=n)(n，k)的一个递推公式，它非常简单，而且对于计算目的很有用(见定理3.1)。我们还注意到Ln和N^之间的一个有趣的关系。最后，在第四节中，我们证明了YLk=o^(^n=k)=1，并得到了Ln的概率母函数和阶乘矩。对于1<；n<；50，给出了当p=1/2时Ln的均值和方差表。
Let Ln be the length of the longest run of successes in n ( 1) independent trials with constant success probability p (0 k) (1 < k ^ n) for any p £ (0, 1). Formulas are also given for P(Ln^k)' and P(Ln=k) (0 < k < n) . Our formulas are given in terms of the multinomial coefficients and in terms of the Fibonacci-type polynomials of order k (see Lemma 2.1, Definition 2.1, and Theorem 2.1). As a corollary to Theorem 2.1, we find two enumeration theorems of Bollinger [2] involving, in his terminology, the number of binary numbers of length n that do not have (or do have) a string of k consecutive ones. We present these results in Section 2. In Section 3, we reconsider the waiting random variable Nk (k ^ 1) , which denotes the number of Bernoulli trials until the occurrence of the k consecutive success, and we state and prove a recursive formula for P(Nk = n) (n k) which is very simple and useful for computational purposes (see Theorem 3.1). We also note an interesting relationship between Ln and N^. Finally, in Section 4, we show that YLk= o ^(^n = k) = 1 and derive the probability generating function and factorial moments of Ln. A table of means and variances of Ln when p = 1/2 is given for 1 < n < 50.