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--- b:head_first_statistics:using_discrete_probability_distributions [2023/10/01 18:59] – [Theorems] hkimscil
+++ b:head_first_statistics:using_discrete_probability_distributions [2023/10/04 10:29] – hkimscil
@@ Line 530: / Line 530: @@
 & = 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 590: @@
 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 616: @@
 \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 624: @@
 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 691: @@
 \end{align}
 ====== e.gs in R  ======
 R에서 이를 살펴보면
 <code>
-m <- 0
+# variance theorem 4-1, 4-2
-v <- 4
+# http://commres.net/wiki/variance_theorem
-n <- 10000
+# need a function, rnorm2
+rnorm2 <- function(n,mean,sd) { mean+sd*scale(rnorm(n)) }
+m <- 50
+v <- 100
+n <- 1000000
 set.seed(1)
 x1 <- rnorm2(n, m, sqrt(v))
@@ Line 719: / Line 722: @@
 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 748: @@
 # var(2*x1) = 2^2 var(X1)
 v.11
 </code>