# 닭의 목아지를 비틀어도 새벽은 온다. - 1979, 김영삼 -

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statistical_review

# Rules for Variance

1. The variance of a constant is zero.
$\Var(c) = 0$
2. Adding a constant value, c to a variable does not change variance (because the expectation increases by the same amount).
$\sigma_{x+c} = Var(X+c) = E[((X_{i} + c)-E(\overline{X} + c))^{2}] = Var(X)$
3. Multiplying a constant value, c to a variable increase the variance by square of the constant, c.
$\sigma_{c*x} = Var(cX) = c^{2}Var(X)$
4. The variance of the sum of two or more random variables is equal to the sum of each of their variances only when the random variables are independent.
$Var(X+Y) = Var(X) + 2 Cov(X,Y) + Var(Y)$
and $Cov(X,Y) = 0$

# Rules for the Covariance

1. The covariance of two constants, c and k, is zero.
$Cov(c,k) = E[(c-E(c))(k-E(k)] = E[(0)(0)] = 0$
2. The covariance of two independent random variables is zero.
$Cov(X, Y) = 0$ When X and Y are independent.
3. The covariance is a combinative as is obvious from the definition.
$Cov(X, Y) = Cov(Y, X)$
4. The covariance of a random variable with a constant is zero.
$Cov(X, c) = 0$
5. Adding a constant to either or both random variables does not change their covariances.
$Cov(X+c, Y+k) = Cov(X, Y)$
6. Multiplying a random variable by a constant multiplies the covariance by that constant.
$Cov(cX, kY) = c*k \: Cov(X, Y)$
7. The additive law of covariance holds that the covariance of a random variable with a sum of random variables is just the sum of the covariances with each of the random variables.
$Cov(X+Y, Z) = Cov(X, Z) + Cov(Y, Z)$
8. The covariance of a variable with itself is the variance of the random variable.
$Cov(X, X) = Var(X)$