[統計]離散型(Discrete)隨機變數與其機率分佈(2)
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
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
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， $$ \begin{align} E[\sin(X)] & = \sum_{\substack{x \in R_X}}\sin(x)P(x) \ & = \sin(0)P_X(0) + \sin(\frac{\pi}{4})P_X(\frac{\pi}{4}) + \sin(\frac{\pi}{2})P_X(\frac{\pi}{2}) + \sin(\frac{3\pi}{4})P_X(\frac{3\pi}{4}) + \sin(\pi)P_X(\pi)\ & = 0 \times \frac{1}{5} + \frac{\sqrt{2}}{2} \times \frac{1}{5} + 1 \times \frac{1}{5} + \frac{\sqrt{2}}{2} \times \frac{1}{5} + 0 \times \frac{1}{5} \ & = \frac{\sqrt{2} + 1}{2} \end{align} $$
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
$$ Var(X) = E[(X - \mu)^2] = \sum_{\substack{x \in R_X}}(x - \mu)^2P(x) $$
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：
$$ \begin{align} \sigma^2 & = E[(X - \mu)^2] & \ & = E[(X^2 - 2\mu X + \mu^2] \ & = E(X^2) - E(2\mu X) + E(\mu^2) \ & = E(X^2) - 2\mu E(X) + \mu^2 \ & = E(X^2) - 2\mu^2 + \mu^2 \ & = E(X^2) - \mu^2 \ & = E(X^2) - E^2(X) \ \end{align} $$
參考：