• 如果要用一個確定的值來代表一個隨機變數，應該要怎麼挑？
• 什麼是隨機變數的平均？
• \(P(X = 1) = P(X = 2) = 0.4, P(X = 3) = P(X = 4) = 0.1\)
  • \(X\)的值平均來說是多少？
• \(X\)重複100次，約40次是1, 40次是2, 10次3, 10次4，那平均是多少？

\[\frac{1}{N} \sum_{x} x \times N \times P(X = x)\]

• \(X\)是一個隨機變數，\(g : \mathbb{R} \rightarrow \mathbb{R}\)是\(X\)的函數，則\(g(X)\)的期望值是：

\[E_X g(X) = \left\{\begin{array}{ll} \int_{-\infty}^{\infty}g(x)f_X(x) dx & \text{ 如果$X$是連續的 } \\ \sum_x g(x) P(X = x) & \text{ 如果$X$是離散的 } \end{array}\right.\]

• \(S\) 代表骰兩次獨立的公平的骰子
• \(X\) 代表點數的和

\[P(X = 2) = \frac{1}{36} , P(X = 3) = \frac{2}{36} , P(X = 4) = \frac{3}{36}\] \[P(X = 5) = \frac{4}{36} , P(X = 6) = \frac{5}{36} , P(X = 7) = \frac{6}{36}\] \[P(X = 8) = \frac{5}{36} , P(X = 9) = \frac{4}{36} , P(X = 10) = \frac{3}{36}\] \[P(X = 11) = \frac{2}{36} , P(X = 12) = \frac{1}{36}\]

• \(E(X) = \sum_{x} P(X = x) x = 7\)

• 如果\(X\)是連續的隨機變數，而且PDF為：

\[f_X(x) = \frac{1}{\pi} \times \frac{1}{1 + x^2}\]

• \(f_X\)有以下性質：
  • \(\int_{-\infty}^{\infty} f_X(x) = 1\)
  • \(\int_{-M}^M x f_X(x) dx = \frac{1}{\pi} log(1 + M^2)\)，所以\(E(X) = \infty\) 不存在

• 若\(g(X) = a\)，則\(E(g(X)) = a\)
• \(E(a g_1(X) + b g_2(X) + c) = a E(g_1(X)) + b E(g_2(X)) + c\)
• 若\(g_1(x) \leq g_2(x) \leq g_3(x)\)，則\(E(g_1(X)) \leq E(g_2(X)) \leq E(g_3(X))\)