這一系列我是以 Wackerley, Mendenhall and Scheaffer 的
Mathematical Statistics with Applications, 7th edition 這本書為主，
有些名詞與定理解釋會再參考其他書籍與網路作為輔助（下方有參考連結）。

這篇關於期望值的介紹在書中的 3.3，但這篇內容有很大部分也來自 Source

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期望值 Expectation

定義 Definition

• Let \(X\) be a discrete random variable with range \(R_X = \{ x_1, x_2, x_3,… \} \)(finite or countably infinite). The expectation of \(X\) is defined as

\[E(X) = \sum_{\substack{x_k \in R_X}}x_kP(X=x_k)=\sum_{\substack{x_k \in R_X}}x_kP_X(x_k)\]

且期望值 \(E(X)\) 會等於平均數 \(\mu\)

\[E(X) = \mu\]

說明：

There is a discrete random variable with range \(R_X = \{ x_1, x_2, x_3,… \} \). Suppose that we repeat this experiment a very large number of times \(N\), and that the trials are independent. Let \(N_1\) be the number of times we observe \(x_1\), \(N_2\) be the number of times we observe \(x_2, ….\), \(N_k\) be the number of times we observe \(x_k\).

\[P_X(x_1) \approx\frac{N_1}{N} \\ P_X(x_2) \approx\frac{N_2}{N} \\ \cdots\\ P_X(x_k) \approx\frac{N_k}{N} \\ \cdots\]

從以上式子我們可以得到 \(N_k = NP_X(x_k)\)

\[\begin{align} \mu & = \frac{N_1x_1 + N_2x_2 + N_3x_3 + ...}{N} \\ & \approx \frac{x_1NP_X(x_1) + x_2NP_X(x_2) + x_3NP_X(x_3) + ...}{N} \\ & = x_1P_X(x_1) + x_2P_X(x_2) + x_3P_X(x_3) + ... \\ & = E(X) \end{align}\]

以上就是 \(E(X) = \mu\) 的說明。

另外，可能會看到一些不同的寫法表示期望值

\[E(X)=E[X]=EX=\mu=\mu_X\]

這部分內容主要來源為 Source

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Law of the unconscious statistician (LOTUS)

THEOREM 3.2

• Let \(X\) be a discrete random variable with probability function \(P(x)\) and \(g(X)\) be a real-valued function of \(Y\). Then the expected value of \(g(X)\) is given by

\[E[g(X)] = \sum_{\substack{x \in R_X}}g(x)P(x)\]

Example:
Source

Let \(X\) be a discrete random variable with range \(R_X = \{ 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi… \} \), such that \(P_X(0)=P_X(\frac{\pi}{4})=P_X(\frac{\pi}{2})=P_X(\frac{3\pi}{4})=P_X(\pi)=\frac{1}{5}\). Find \(E[sin(X)]\).

Solution:

根據 LOTUS， \(\begin{align} E[\sin(X)] & = \sum_{\substack{x \in R_X}}\sin(x)P(x) \\ & = \sin(0)P_X(0) + \sin(\frac{\pi}{4})P_X(\frac{\pi}{4}) + \sin(\frac{\pi}{2})P_X(\frac{\pi}{2}) + \sin(\frac{3\pi}{4})P_X(\frac{3\pi}{4}) + \sin(\pi)P_X(\pi)\\ & = 0 \times \frac{1}{5} + \frac{\sqrt{2}}{2} \times \frac{1}{5} + 1 \times \frac{1}{5} + \frac{\sqrt{2}}{2} \times \frac{1}{5} + 0 \times \frac{1}{5} \\ & = \frac{\sqrt{2} + 1}{2} \end{align}\)

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變異數 Variance

定義 Definition

• If \(X\) is a random variable with mean \(E[X] = \mu\), the variance of \(X\) is defined as

\[Var(X) = E[(X - \mu)^2]\]

• 且根據 LOTUS，這個式子可以再寫成

\[Var(X) = E[(X - \mu)^2] = \sum_{\substack{x \in R_X}}(x - \mu)^2P(x)\]

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期望值的運算

THEOREM 3.3

• Let X be a discrete random variable with probability function \(p(x)\) and \(c\) be a constant. Then

\[E(c) = c.\]

THEOREM 3.4

• Let X be a discrete random variable with probability function \(p(x)\) and \(g(X)\) be a function of X, and c be as constant. Then

\[E[cg(X)] = cE[g(X)].\]

THEOREM 3.5

• Let X be a discrete random variable with probability function \(p(x)\) and \(g_1(X), g_2(X), …, g_k(X)\) be \(k\) functions of \(X\). Then

\[E[g_1(X)+g_2(X)+...+g_k(X)] = E[g_1(X)] + E[g_2(X)] + ... + E[g_k(X)]\]

根據以上定理，可以再產生

THEOREM 3.6

• Let X be a discrete random variable with probability function \(p(x)\) and mean \(E(X) = \mu\). Then

\[\begin{align} V(X) = \sigma^2 & = E[(X - \mu)^2] \\ & = E(X^2) - \mu^2 \\ & = E(X^2) - E^2(X) \\ \end{align}\]

證明：

\[\begin{align} \sigma^2 & = E[(X - \mu)^2] & \\ & = E[(X^2 - 2\mu X + \mu^2] \\ & = E(X^2) - E(2\mu X) + E(\mu^2) \\ & = E(X^2) - 2\mu E(X) + \mu^2 \\ & = E(X^2) - 2\mu^2 + \mu^2 \\ & = E(X^2) - \mu^2 \\ & = E(X^2) - E^2(X) \\ \end{align}\]

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參考：

• Mathematical Statistics with Applications 7th Edition
• Introduce To Proability, Statistics and Random Process
• 陳鍾誠的網站