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--- expected_value_and_variance_properties [2023/10/04 12:42] – [Theorems] hkimscil
+++ expected_value_and_variance_properties [2023/12/07 12:18] (current) – [Theorem 2: Why square] hkimscil
@@ Line 9: / Line 9: @@
 | $Var(c)$  | $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 $\ref{var.theorem.2} and \ref{var.theorem.52}$ |
+| $Var(aX - bY)$ | $a^{2}Var(X) + b^{2}Var(Y)$ see $\ref{var.theorem.2}$ and $\ref{var.theorem.52}$ |
 | $Var(X1 + X2 + X3)$ | $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 33: / Line 33: @@
 \begin{align*}
-Var[aX] & = & E[a^2X^2] − (E[aX])^2 \\
+Var[aX] & = E[a^2X^2] − (E[aX])^2 \\
- & = & a^2 E[X^2] - (a E[X])^2 \\
+ & = a^2 E[X^2] - (a E[X])^2 \\
- & = & a^2 E[X^2] - (a^2 E[X]^2) \\
+ & = a^2 E[X^2] - (a^2 E[X]^2) \\
- & = & a^2 (E[X^2] - (E[X])^2) \\
+ & = a^2 (E[X^2] - (E[X])^2) \\
- & = & a^2 (Var[X]) \label{var.theorem.2} \tag{variance theorem 2} \\
+ & = a^2 (Var[X]) \label{var.theorem.2} \tag{variance theorem 2} \\
 \end{align*}
 ====== Theorem 3: Why Var[X+c] = Var[X] ======
@@ Line 153: / Line 153: @@
 보통 X1, X2 집합은 같은 특성을 (statistic) 갖는 두 독립적인 집합을 의미하므로
 \begin{align*}
-Var(X1 + X2)  = Var(X1) + Var(X2)  \\
+Var(X1 + X2) = & Var(X1) + Var(X2)  \\
+ & \text{because X1 and x2 have} \\
+ & \text{X's statistics (the same mean} \\
+ & \text{and variance of X)} \\
+= & Var(X) + Var(X) \\
 \end{align*}
@@ Line 173: / Line 178: @@
 & \;\;\;\;\; \text{according to the below } \ref{cov.xx}, \\
 & \;\;\;\;\; Cov(X,X) = Var(X) \\
-& = Var(X) + 2 Var(X) + Var(X)   \\
+& = Var(X) + 2 Var(X) + Var(X) \;\;\; \\
 & = 4 Var(X)
 \end{align*}
@@ Line 191: / Line 196: @@
 rnorm2 <- function(n,mean,sd) { mean+sd*scale(rnorm(n)) }
-m <- 0
+m <- 50
-v <- 1
+v <- 4
-n <- 10000
+n <- 100000
 set.seed(1)
 x1 <- rnorm2(n, m, sqrt(v))
@@ Line 199: / Line 204: @@
 x3 <- rnorm2(n, m, sqrt(v))
-m.x1 <- mean(x1)
+m.x1 <- round(mean(x1),3)
-m.x2 <- mean(x2)
+m.x2 <- round(mean(x2),3)
-m.x3 <- mean(x3)
+m.x3 <- round(mean(x3),3)
 m.x1
 m.x2
 m.x3
+y1 <- 3*x1 +5
+exp.y1 <- mean(y1)
+exp.3xplus5 <- 3 * mean(x1) + 5
+exp.y1
+exp.3xplus5
 v.x1 <- var(x1)
@@ Line 212: / Line 223: @@
 v.x2
 v.x3
+var(x1)
+var((3*x1)+5)
+^2 * var(x1)
 v.12 <- var(x1 + x2)