b:head_first_statistics:using_discrete_probability_distributions
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b:head_first_statistics:using_discrete_probability_distributions [2023/10/01 12:58] – [Theorems] hkimscil | b:head_first_statistics:using_discrete_probability_distributions [2023/10/04 10:29] – hkimscil | ||
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====== Theorems ====== | ====== Theorems ====== | ||
- | | E(X< | + | | $E(X)$ | $\sum{X}\cdot P(X=x)$ |
- | | E(aX + b) | $aE(X) + b$ | | + | | $E(X^2)$ | $\sum{X^{2}}\cdot P(X=x)$ | |
- | | Var(aX + b) | $a^{2}Var(X) \; | + | | $E(aX + b)$ | $aE(X) + b$ | |
- | | E(X) | $\sum{x} \cdot P(X=x) $ | | + | | $E(f(X))$ | $\sum{f(X)} \cdot P(X=x)$ |
- | | E(f(X)) | $\sum{f(X)} \cdot P(X=x)$ | | + | | $E(aX - bY)$ | $aE(X)-bE(Y)$ | |
- | | Var(aX - bY) | $a^{2}Var(X) + b^{2}Var(Y)$ see 1 | | + | | $E(X1 + X2 + X3)$ | $E(X) + E(X) + E(X) = 3E(X) \; |
- | | Var(X) | $E(X-\mu)^{2} = E(X^{2})-\mu^{2} \;\;\; $ see $\ref{var.theorem.1} $ | | + | | $Var(X)$ | $E(X-\mu)^{2} = E(X^{2})-E(X)^{2} \;\;\; $ see $\ref{var.theorem.1} $ | |
- | | E(aX - bY) | $aE(X)-bE(Y)$ | | + | | $Var(c)$ |
- | | E(X< | + | | $Var(aX + b)$ | $a^{2}Var(X) \; |
- | | 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, | + | | $Var(aX - bY)$ | $a^{2}Var(X) + b^{2}Var(Y)$ see 1 | |
- | | Var(X1 + X1 + X1) | $Var(3X) = 3^2 Var(X) = 9 Var(X) $ | | + | | $Var(X1 + X2 + X3)$ | $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, |
+ | | $Var(X1 + X1 + X1)$ | $Var(3X) = 3^2 Var(X) = 9 Var(X) $ | | ||
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& = E[X^2] - 2 \mu^2 + \mu^2 | & = E[X^2] - 2 \mu^2 + \mu^2 | ||
& = E[X^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} | \end{align} | ||
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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 | ||
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\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)) | ||
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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 | ||
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# 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