expected_value_and_variance_properties
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| expected_value_and_variance_properties [2023/10/04 12:58] – [e.gs in R] hkimscil | expected_value_and_variance_properties [2023/12/07 12:18] (current) – [Theorem 2: Why square] hkimscil | ||
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| \begin{align*} | \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*} | \end{align*} | ||
| ====== Theorem 3: Why Var[X+c] = Var[X] ====== | ====== Theorem 3: Why Var[X+c] = Var[X] ====== | ||
| Line 196: | Line 196: | ||
| rnorm2 <- function(n, | rnorm2 <- function(n, | ||
| - | m <- 0 | + | m <- 50 |
| v <- 4 | v <- 4 | ||
| n <- 100000 | n <- 100000 | ||
| Line 210: | Line 210: | ||
| m.x2 | m.x2 | ||
| m.x3 | m.x3 | ||
| + | |||
| + | y1 <- 3*x1 +5 | ||
| + | exp.y1 <- mean(y1) | ||
| + | exp.3xplus5 <- 3 * mean(x1) + 5 | ||
| + | exp.y1 | ||
| + | exp.3xplus5 | ||
| v.x1 <- var(x1) | v.x1 <- var(x1) | ||
| Line 217: | Line 223: | ||
| v.x2 | v.x2 | ||
| v.x3 | v.x3 | ||
| + | |||
| + | var(x1) | ||
| + | var((3*x1)+5) | ||
| + | 3^2 * var(x1) | ||
| v.12 <- var(x1 + x2) | v.12 <- var(x1 + x2) | ||
expected_value_and_variance_properties.1696391896.txt.gz · Last modified: 2023/10/04 12:58 by hkimscil