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Rules for Variance
The variance of a constant is zero.
Adding a constant value, c to a variable does not change variance (because the expectation increases by the same amount).
σx+c=Var(X+c)=E[((Xi+c)−E(¯X+c))2]=Var(X)
Multiplying a constant value, c to a variable increase the variance by square of the constant, c.
σc∗x=Var(cX)=c2Var(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)+2Cov(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
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∗kCov(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)