本研究以多元序列為題材，分別對序列中的連個數和最長的連提出其精確的分佈，其中多元序列指的是至少由3 種元素所構成的序列，如DNA序列(腺嘌呤(A)、胞嘧啶(C)、鳥嘌呤(G)和胸腺嘧啶(T))。序列中的連常用來評斷該序列的出現是否為隨機，在各元素出現機率相等的情況下，若連個數太多或太少皆可懷疑該序列的出現並非隨機的。此類問題在二元序列中已有廣泛的研究，而本文將以DNA 序列為例，分別先定義A、T、C、G 生成的機率，再利用FMCI 之技巧，分別對元素取後放回與取後不放回提出精確的分佈，最後探討這兩個假設在計算上的效率，另外也會額外探討二元序列在元素取後不放回的假設下，其最長的連和連個數的精確分佈。

Number of runs and longest of run for a multi-state
sequence are studied in this thesis. Our target in research is to find the exact distribution. We will use the DNA sequence to be the example. The possible elements are A, C, G, and T, representing the four nucleotide bases
of a DNA strand. We define the generating probability of the four elements first, then we use FMCI to provide the exact distribution for element with replacement and element without replacement respectively, and finally exploring the efficiency of two methods. Also , we will conduct the exact distribution of run for element without replacement in a two-state sequence.

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