b:head_first_statistics:using_discrete_probability_distributions
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b:head_first_statistics:using_discrete_probability_distributions [2023/10/01 19:00] – [Variance Theorem 1] hkimscil | b:head_first_statistics:using_discrete_probability_distributions [2023/10/04 10:29] – hkimscil | ||
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Variance는 기본적으로 아래와 같다. 이 때 $X=c$ 라고 (c=상수) 하면 | Variance는 기본적으로 아래와 같다. 이 때 $X=c$ 라고 (c=상수) 하면 | ||
\begin{align} | \begin{align} | ||
- | Var(X) | + | Var(X) & = E[(X − E(X))^2] \text{ |
- | & = E[(X − E(X))^2] \text{ | + | |
& = E[(c-c)^2] \nonumber | & = E[(c-c)^2] \nonumber | ||
& = 0 | & = 0 | ||
Line 617: | Line 616: | ||
\begin{align*} | \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^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*} | \end{align*} | ||
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Var[(X+Y)] = | Var[(X+Y)] = | ||
& E[X^2] & + & 2E[XY] & + & E[Y^2] \\ | & 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] \\ | & Var[X] & + & 2 E[XY]-2E(X)E(Y) & + & Var[Y] \\ | ||
\end{align*} | \end{align*} | ||
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\end{align} | \end{align} | ||
====== e.gs in R ====== | ====== e.gs in R ====== | ||
- | |||
R에서 이를 살펴보면 | R에서 이를 살펴보면 | ||
< | < | ||
- | m <- 0 | + | # variance theorem 4-1, 4-2 |
- | v <- 4 | + | # http:// |
- | n <- 10000 | + | # need a function, rnorm2 |
+ | rnorm2 <- function(n, | ||
+ | |||
+ | m <- 50 | ||
+ | v <- 100 | ||
+ | n <- 1000000 | ||
set.seed(1) | set.seed(1) | ||
x1 <- rnorm2(n, m, sqrt(v)) | x1 <- rnorm2(n, m, sqrt(v)) | ||
Line 719: | Line 722: | ||
v.12 <- var(x1 + x2) | v.12 <- var(x1 + x2) | ||
v.12 | v.12 | ||
- | ## should be near 2*v | + | ###################################### |
+ | ## v.12 should be near var(x1)+var(x2) | ||
###################################### | ###################################### | ||
## 정확히 2*v가 아닌 이유는 x1, x2가 | ## 정확히 2*v가 아닌 이유는 x1, x2가 | ||
Line 744: | Line 748: | ||
# var(2*x1) = 2^2 var(X1) | # var(2*x1) = 2^2 var(X1) | ||
v.11 | v.11 | ||
- | |||
</ | </ |
b/head_first_statistics/using_discrete_probability_distributions.txt · Last modified: 2023/10/04 10:29 by hkimscil