臺灣博碩士論文加值系統

在研究連續型隨機變數的機率密度函數的行為時，如果函數是可微分或是片段可微微分，通常可以應用微積分的方法去判斷函數圖形的增減性、凹性、反曲點及漸進線以以獲得機率分配的某些特性。這些被使用到的微積分工具，大多是導數均值定理的推論論。該定理要求所給定的函數在一有限閉區間上連續且在該區間內部可微，一般而言連連續的機率密度函數未必是處處可微的，典型的例子如片段線性的連續分配與雙重指數數分配的機率密度函數等，但是在上述的例子中不可微分的地方在左導數與右導數都存存在。
本文中我們將考慮一個定義在實數系中的某個區間上的單邊可微分的連續數，以基本的方法證明一個單邊可微分函數的均值定理，我們也討論這個結果在函數增減性與凹性的應用，並且以一些機率密度函數加以驗證上述的結論。

In the study of the behavior of probability density function of continuous random variable, if the functions are differentiable or piecewise differentiable, usually, one can apply the method of calculus to determine the monotonically, concavity, points of inflection and asymptotes of these functions to attain some properties of the probability distributions. Most of the tools in calculus are consequences of the Mean Value Theorem for Derivatives. It is a theorem about functions continuous in bounded closed intervals and differentiable in the interior of the intervals. In general, continuous probability density functions are not necessarily differentiable everywhere, typical examples such as continuous piecewise linear distributions and double exponential distributions, but they have both left and right derivatives at the points where they are not differentiable.
In this thesis, we shall consider one-sided differential functions defined on some intervals in the real number system and attain a Mean Value Theorem for One-sided Derivatives by an elementary proof. We also apply the result to discuss the monotonically and concavity of functions by examine some probability density functions.

目 錄
頁次
第一章 緒論 1
第一節 研究動機與目的 1
第二節 論文架構 3
第二章 單邊可微分函數的均值定理 4
第一節 預備定理 4
第二節 單邊導數的均值定理 13
第三章 應用 20
第一節 單邊可微函數增減性 20
第二節 單邊可微分性及凸性 33
第四章 結論 38
參考文獻 39

參考文獻
(1)Berkey, D. D., and Blanchard, P. calculus, 3nded. Saunders College Publishing. 2000.
(2)Larson, H. J. Introduction to Probability Theorem and Statistical Inference, 3nded. 1962.
(3)Mcleod R., Mean value theorems for vector valued functions, Proc. Edinburgh Math. Soc. (2)(1964-65)
(4)Rice, J. R. Mathematical Statistics and Data analysis, 2nded. Duxbury Press, 1995.
(5)Ross, S. M. A First Course in Probability, 2nded. Macmillam Publishing Company, 1984.