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--- statistical_review [2017/12/11 09:16] – [Rules for the Covariance] hkimscil
+++ statistical_review [2023/10/05 17:30] (current) – [Rules for the Covariance] hkimscil
@@ Line 1: / Line 1: @@
 ====== Rules for Variance ======
-  - The variance of a constant is zero. \\ $\Var(c) = 0 $
+see [[:expected value and variance properties]]
-  - 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 $
 ====== Rules for the Covariance ======
-  - The covariance of two constants, c and k, is zero. \\ $Cov(c,k) = E[(c-E(c))(k-E(k)] = E[(0)(0)] = 0$
+see [[:covariance properties]]
-  - 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) $