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--- b:head_first_statistics:using_discrete_probability_distributions [2023/10/01 13:07] – [Theorems] hkimscil
+++ b:head_first_statistics:using_discrete_probability_distributions [2024/10/01 21:44] (current) – [Fat Dan changed his prices] hkimscil
@@ Line 281: / Line 281: @@
 __Pool puzzle__
+\begin{eqnarray*}
+\text {original win} & = & \text {c(0, 5, 10, 15, 20)} \\
+\text {original cost} & = & \text {c(1)} \\
+\text {X} & = & \text {the amount of money you receive} \\
+\end{eqnarray*}
 \begin{eqnarray*}
@@ Line 295: / Line 301: @@
 E(X) = -.77 and E(Y) = -.85. What is 5 * E(X) + 3?
-$ 5 * E(X) = -3.85 $
+\begin{eqnarray*}
-$ 5 * E(X) + 3 = -0.85 $
+E(X) & = & -.77 \\
-$ E(Y) = 5 * E(X) + 3 $
+E(Y) & = & E(5X+3) \;\;\;\;\;  \because Y=5X+3 \\
+& = & E(5X) + 3 \\
+& = & 5 E(X) + 3 \\
+& = & -0.85 \\
+\\
+\\
+Var(Y) & = & Var(5X+3) \\
+& = & Var(5X) + Var(3) \\
+& = & Var(5X) \\
+& = & 5^2 Var(X) \\
+& = & 67.4275
+\end{eqnarray*}
-$ 5 * Var(X) = 13.4855 $
-$ 5^2 * Var(X) = 67.4275 $
-$ Var(Y) = 5^2 * Var(X) $
 \begin{eqnarray*}
@@ Line 375: / Line 388: @@
 </code>
-그런데
+위는 각각 계산한 것. 아래는 Y = X - 0.5를 적용해서 계산한 것
 \begin{eqnarray*}
 E(X-a) & = & E(X) - a \\
@@ Line 507: / Line 520: @@
 | $E(aX - bY)$ | $aE(X)-bE(Y)$  |
 | $E(X1 + X2 + X3)$ | $E(X) + E(X) + E(X) = 3E(X) \;\;\; $ ((X1,X2,X3는 동일한 statistics을 갖는 (X의 특성을 갖는, 즉, 집합 X의 동일한 mean, variance, sdev 값을 갖는) 집합))   |
+| $Var(X)$ | $E(X-\mu)^{2} = E(X^{2})-E(X)^{2} \;\;\; $   see $\ref{var.theorem.1} $ |
 | $Var(c)$  | $0 \;\;\; $ see $\ref{var.theorem.41}$   |
 | $Var(aX + b)$ | $a^{2}Var(X) \;\;\; $  see $\ref{var.theorem.2}$ and $\ref{var.theorem.3}$ |
 | $Var(aX - bY)$ | $a^{2}Var(X) + b^{2}Var(Y)$ see 1 |
-| $Var(X)$ | $E(X-\mu)^{2} = E(X^{2})-\mu^{2} \;\;\; $   see $\ref{var.theorem.1} $ |
 | $Var(X1 + X2 + X3)$ | $Var(X) + Var(X) + Var(X) = 3 Var(X) \;\;\; $ ((X1, x2, x3는 동일한 특성을 (statistic, 가령 Xbar = 0, sd=1) 갖는 독립적인 세 집합이다. 따라서 세집합의 분산은 모두 1인 상태이고, 이들의 분삽값은 모두 동일하므로 Var(3X)의 성질을 갖는다.))  |
 | $Var(X1 + X1 + X1)$  | $Var(3X) = 3^2 Var(X) = 9 Var(X) $  |
@@ Line 530: / Line 543: @@
 & = E[X^2] - 2 \mu^2 + \mu^2   \nonumber \\
 & = E[X^2] - \mu^2 \nonumber \\
-& = E[X^2] - (E[X])^2 \label{var.theorem.1} \tag{variance theorem 1} \\
+& = E[X^2] - E[X]^2 \label{var.theorem.1} \tag{variance theorem 1} \\
 \end{align}
@@ Line 590: / Line 603: @@
 Variance는 기본적으로 아래와 같다. 이 때 $X=c$ 라고 (c=상수) 하면
 \begin{align}
-Var(X) & = E[(X − E(X))^2] \nonumber \\
+Var(X) & = E[(X − E(X))^2] \text{    because  X = c, and E(X) = c}    \nonumber \\
-& = E[(X − E(X))^2] \text{    because  X = c, and E(X) = c}    \nonumber \\
 & = E[(c-c)^2] \nonumber  \\
 & = 0   \nonumber  \\
@@ Line 617: / Line 629: @@
 \begin{align*}
   &  E[(X + Y)^2] = E[X^2 + 2XY + Y^2] = E[X^2] + 2E[XY] + E[Y^2] \\
-- & [E(X + Y)]^2 = [E(X) + E(Y)]^2 = E(X)^2 + 2E(X)E(Y) + E(Y^2) \\
+- & [E(X + Y)]^2 = [E(X) + E(Y)]^2 = E(X)^2 + 2E(X)E(Y) + E(Y)^2 \\
 \end{align*}
@@ Line 625: / Line 637: @@
 Var[(X+Y)] =
   & E[X^2] & + & 2E[XY] & + & E[Y^2] \\
-- & E(X)^2 & - & 2E(X)E(Y) & - & E(Y^2) \\
+- & E(X)^2 & - & 2E(X)E(Y) & - & E(Y)^2 \\
   & Var[X] & + & 2 E[XY]-2E(X)E(Y) & + & Var[Y] \\
 \end{align*}
@@ Line 692: / Line 704: @@
 \end{align}
 ====== e.gs in R  ======
 R에서 이를 살펴보면
 <code>
+# variance theorem 4-1, 4-2
+# http://commres.net/wiki/variance_theorem
+# need a function, rnorm2
+rnorm2 <- function(n,mean,sd) { mean+sd*scale(rnorm(n)) }
 m <- 0
-v <- 4
+v <- 1
 n <- 10000
 set.seed(1)
@@ Line 719: / Line 735: @@
 v.12 <- var(x1 + x2)
 v.12
-## should be near 2*v
+######################################
+## v.12 should be near var(x1)+var(x2)
 ######################################
 ## 정확히 2*v가 아닌 이유는 x1, x2가
@@ Line 744: / Line 761: @@
 # var(2*x1) = 2^2 var(X1)
 v.11
 </code>