statistical_review

- The variance of a constant is zero.

$\Var(c) = 0 $ - 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) $ - 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)$ - 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 $

- The covariance of two constants, c and k, is zero.

$Cov(c,k) = E[(c-E(c))(k-E(k)] = E[(0)(0)] = 0$ - The covariance of two independent random variables is zero.

$Cov(X, Y) = 0$ When X and Y are independent. - The covariance is a combinative as is obvious from the definition.

$Cov(X, Y) = Cov(Y, X)$ - The covariance of a random variable with a constant is zero.

$Cov(X, c) = 0 $ - Adding a constant to either or both random variables does not change their covariances.

$Cov(X+c, Y+k) = Cov(X, Y)$ - Multiplying a random variable by a constant multiplies the covariance by that constant.

$Cov(cX, kY) = c*k \: Cov(X, Y)$ - 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.

$Cov(X+Y, Z) = Cov(X, Z) + Cov(Y, Z)$ - The covariance of a variable with itself is the variance of the random variable.

$Cov(X, X) = Var(X) $

statistical_review.txt · Last modified: 2017/12/11 09:16 by hkimscil