Let $X$ and $Y$ be independent random variables and $a$ and $b$ are constants. For the sum formulas, $X_1,\ldots,X_n$ are independent copies of $X$, with the same distribution and finite mean and variance. $\text{E}\hspace{0.1em}(a) = a$ $\text{E}\hspace{0.1em}(aX) = a\text{E}\hspace{0.1em}(X)$ $\text{E}\hspace{0.1em}(aX \pm b) = a\text{E}\hspace{0.1em}(X) \pm b$ $\text{E}\hspace{0.1em}(aX \pm bY) = a\text{E}\hspace{0.1em}(X) \pm b\text{E}\hspace{0.1em}(Y)$ $\text{E}\hspace{0.1em}(X_1 + X_2 + \dots + X_n) = n\text{E}\hspace{0.1em}(X)$ $\text{Var}\hspace{0.1em}(a) = 0$ $\text{Var}\hspace{0.1em}(aX) = a^2\text{Var}\hspace{0.1em}(X)$ $\text{Var}\hspace{0.1em}(aX \pm b) = a^2\text{Var}\hspace{0.1em}(X)$ $\text{Var}\hspace{0.1em}(aX \pm bY) = a^2\text{Var}\hspace{0.1em}(X) + b^2\text{Var}\hspace{0.1em}(Y)$ $\text{Var}\hspace{0.1em}(X_1 + X_2 + \dots + X_n) = n\text{Var}\hspace{0.1em}(X)$

Combining Random Variables

Let $X$ and $Y$ be independent random variables and $a$ and $b$ are constants. For the sum formulas, $X_1,\ldots,X_n$ are independent copies of $X$, with the same distribution and finite mean and variance.

$\text{E}\hspace{0.1em}(a) = a$
$\text{E}\hspace{0.1em}(aX) = a\text{E}\hspace{0.1em}(X)$
$\text{E}\hspace{0.1em}(aX \pm b) = a\text{E}\hspace{0.1em}(X) \pm b$
$\text{E}\hspace{0.1em}(aX \pm bY) = a\text{E}\hspace{0.1em}(X) \pm b\text{E}\hspace{0.1em}(Y)$
$\text{E}\hspace{0.1em}(X_1 + X_2 + \dots + X_n) = n\text{E}\hspace{0.1em}(X)$

$\text{Var}\hspace{0.1em}(a) = 0$
$\text{Var}\hspace{0.1em}(aX) = a^2\text{Var}\hspace{0.1em}(X)$
$\text{Var}\hspace{0.1em}(aX \pm b) = a^2\text{Var}\hspace{0.1em}(X)$
$\text{Var}\hspace{0.1em}(aX \pm bY) = a^2\text{Var}\hspace{0.1em}(X) + b^2\text{Var}\hspace{0.1em}(Y)$
$\text{Var}\hspace{0.1em}(X_1 + X_2 + \dots + X_n) = n\text{Var}\hspace{0.1em}(X)$

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