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Variance and Covariance 13 114 Variance and Covariance The central terms are the variance and the covariance. While the variance is interpreted as an 'index of spreading of a random variable' the covariance can be regarded as an 'index for linear connection of random variables'. For the denition of the variance as an index of spreading of a random variable, the squared distances between the values of the random variable and its expectation weighted by the presupposed probability measure has much to be recommended. 13.1 Denition (Variance) Let (Ω, A, P ) be a probability space and X : (Ω, A, P ) → (R, B) a random variable with expectation E(X). Then the value (13.1.1) V (X) := VP (X) := EP ((X − E(X))2 , is called the variance of X w.r.t. P , if it is nite. 13.2 Remarks The variance has been introduced as the expectation of the squared deviation of a random variable X from Variance and Covariance 115 its expectation E(X). Accordingly, the variance is an index for the average quadratic distance of the random variable X from its expectation. A large variance means thereby a large spreading, a small variance a small spreading of the values X(ω), ω ∈ Ω, around the expectation. This is conrmed by the fact that V (X) = 0 is equivalent with the validity of X = E(X) on the whole Ω with the exception of maximally a P nullset. 13.3 Calculation rules for variances Let (Ω, A, P ) be a probability space and let the random variable X : (Ω, A, P ) → (R, B) have the variance V (X). Then 13.3.1 V (aX + b) = a2 V (X), 13.3.2 V (X) = E(X 2 )−[E(X)]2 (a, b ∈ R) (Shifting theorem) Proof: 13.3.1: Due to the linearity of the expectation, we have V (aX+b) = E((aX−E(aX))2 ) = E(a2 (X−E(X))2 = a2 V (X) . 13.3.2: Applying the calculation rules for expectation, Variance and Covariance 116 cf. 12.4 we obtain V (X) = E((X − E(X))2 ) = E(X 2 − 2XE(X) + (E(X))2 ) = E(X 2 ) − 2E(X)E(X) − (E(X))2 = E(X 2 ) − (E(X))2 . The following list contains the variances of particular distributions. Analogously to 12.8 there for expectations we understand under the variance of B(n, p), for example, the variance of VB(n,p) ( idN0 ) of the random variable idN0n w.r.t. the distribution (probability measure) B(n, p) etc. 13.4 Variances of particular distributions 13.4.1 VB(1,p) ( id{0,1} ) = p(1 − p) 13.4.2 VB(n,p) ( idN0n ) = np(1 − p) 13.4.3 Vπ(λ) ( idN ) = λ 13.4.4 VN(a,σ2 ) ( idR ) = σ2 . (We renounce here consciously to present calculation techniques!) 13.5 Denition Let (Ω, A, P ) be a probability space and X, Y : (Ω, A, P ) → (R, B) two random variables whose variances exist. Variance and Covariance 117 Then the value Cov(X, Y ) := Cov(X, Y ) := E((X−E(X))(Y −EY )) is called the covariance of X and Y w.r.t. P . Theorem 13.6 shows in what way the term covariance enters into the theory presented hitherto. 13.6 Theorem Let (Ω, A, P ) be a probability space and X, Y two real random variables with the variances V (X) and V (Y ) respectively. Then the random variable X + Y has also a variance, and V (X + Y ) = V (X) + V (Y ) + 2(X, Y ) . 13.7 Calculation rules for covariances Let X, Y be two random variables with the variances V (X) and V (Y ). Then 13.7.1 Cov(aX+c, bY +d) = abCov(X, Y ) a, b, c, d ∈ R) 13.7.2 Cov(X, Y ) = E(X, Y ) − E(X)E(Y ) Variance and Covariance If moreover X and Y are dent, then 118 stochastically indepen- 13.7.3 (X, Y ) = 0 13.7.4 V (X + Y ) = V (X) + V (Y ) mula of Bienaymé) (For- Hints concerning the proof: 13.7.1 and 13.7.2 correspond to the facts 13.3.1 and 13.3.2 (shifting theorem) respectively formulated for variances. 13.7.3 follows from 13.7.2 and Theorem 12.9 while 13.7.4 is a consequence of Theorem 13.6 and 13.7.3. 13.8 Remark The covariance Cov(X, Y ) is a 'measure' of the linear connection of the random variables X and Y ; the stochastic independence of X and Y implies that Cov(X, Y ) = 0, cf. 13.7.3; the reversal of this fact does not hold. However, in the case of stochastic independence the variance of the sum of two random variables turns out to be simply the sum of the variances, cf. 13.7.4, (Formula of Bienaymé).