Differences

This shows you the differences between two versions of the page.

--- expected_value_and_variance_properties [2023/10/04 12:58] – [e.gs in R] hkimscil
+++ expected_value_and_variance_properties [2025/09/06 10:28] (current) – [e.gs in R] hkimscil
@@ 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 193: / Line 193: @@
 # 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)) }
+rnorm2 <- function(n,mean,sd) {
+    mean+sd*scale(rnorm(n))
+}
-m <- 0
+m <- 50
 v <- 4
 n <- 100000
@@ Line 204: / Line 207: @@
 x3 <- rnorm2(n, m, sqrt(v))
-m.x1 <- round(mean(x1),3)
+# Note: x1, x2, x3는 평균과 표준편차를
-m.x2 <- round(mean(x2),3)
+# 같은 값으로 갖는 (공유하는) 독립적인
-m.x3 <- round(mean(x3),3)
+# 집단
-m.x1
-m.x2
-m.x3
-v.x1 <- var(x1)
+y1 <- 3*x1 +5
-v.x2 <- var(x2)
+exp.y1 <- mean(y1)
-v.x3 <- var(x3)
+exp.3xplus5 <- 3 * mean(x1) + 5
-v.x1
+exp.y1
-v.x2
+exp.3xplus5
-v.x3
+var(x1)
+var((3*x1)+5)
+^2 * var(x1)
+var(y1)  # 9 * var(x) 위와 동일
 v.12 <- var(x1 + x2)
 v.12
 ######################################
 ## v.12 should be near var(x1)+var(x2)
 ######################################
-## 정확히 2*v가 아닌 이유는 x1, x2가
+# 정확히 2*v가 아닌 이유는 x1, x2가
-## 아주 약간은 (random하게) dependent하기 때문
+# 아주 약간은 (random하게) dependent하기 때문
-##(혹은 상관관계가 있기 때문)
+# (혹은 상관관계가 있기 때문, covariance가
-## theorem 5-1 에서
+# 있기 때문)
-## var(x1+x2) = var(x1)+var(x2)+ (2*cov(x1,x2))
+# theorem 5-1 에서
+# var(x1+x2) = var(x1)+var(x2)+ (2*cov(x1,x2))
 cov.x1x2 <- cov(x1,x2)
@@ Line 245: / Line 251: @@
 v.11 <- var(x1 + x1)
 # var(2*x1) = 2^2 var(X1)
+^2*var(x1)
 v.11
 </code>