傳統統計學建立在隨機性上，具有成熟的理論背景與架構，但當隨機發生的事件具有模糊現象時，統計學能針對隨機現象提供完善的解決方法，卻難以給予模糊現象適當的解決方式和解釋，故有模糊隨機變數相關研究的出現，期望在隨機實驗下，當實驗結果是具實數值之的模糊觀測值或本質上會轉化成模糊數時，能透過模糊隨機變數提供合適的分析工具，以便延伸統計學的方法，應用在具有隨機性與模糊性的現象。
本研究以距離測度的概念建立推導模糊觀測資料之期望值與變異數，透過公式之推導分離資料的隨機性和模糊性兩部份，使得模糊型態資料的集中趨勢和離散趨勢得以顯現，可藉統計學的相關性質來描述或推論此模糊資料；本研究建立的期望值與變異數公式可保留資料隱含的原始資訊，除了表達模糊性外，亦符合傳統統計學的定義，以作為統計學的擴充形式，彌補在模糊觀測資料上應用的不足，並分別從端點和變異數為出發點延伸至統計製程管制中管制圖資料的應用，針對平均數與標準差管制圖額外建立模糊性管制界限，較解模糊化方法保留較多的原始資訊，亦無需額外計算模糊資料超出管制界限的比例，即可偵測來自人為判斷的異常現象。最後透過案例演算與分析，顯示本研究方法之變異數與所建立之管制圖具有較佳的偵測能力及合理性。

Randomness is the foundation of the traditional statistics which consists in mature theoretical background. Statistics are able to provide appropriate solutions when events appear randomly. However, if the events are classified as fuzzy phenomena, it is difficult to offer a suitable method and explanation for the vague characteristic of the events. Therefore, the relative study of fuzzy random variable has been developed in order to provide proper analysis tools for random experiments for which the outcome is based upon fuzzy observations with real values or turn to a fuzzy number in nature, so that it can extend the method of statistics to apply to other situations characterized by both randomness and fuzziness.
This study constructs the expected value and variance of fuzzy observations using the concept of distance metrics. The randomness and fuzziness is separated from the data through deriving formulations which demonstrate the central tendency and the dispersal tendency of the vague data, and then describe or infer them though relative properties based upon statistics. The expected value and variance that this study constructs can retain the original information which is implicit in the data. It is not only able to express the characteristics of fuzziness but also conform to the definition of traditional statistic through deriving the expected value and variance. As an extension of statistics, the expected value and variance of fuzzy random variables can make up the application deficiency which results from fuzzy observations. For the application to the control chart in statistical process control, the construction of fuzzy control limits of the chart through the basis of the endpoint and the variance of the fuzzy numbers can detect an unusual status which results human behavior or subjective judgment and thus saves more original information without additionally calculating the proportion vague data which fall outside of the control limit. Furthermore, this study illustrates that the proposed variance and the established control limits have better detective ability and reasonableness by analyzing the example.
Key words: Fuzzy random variable, expected value, variance, fuzzy control chart.

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