Covariance

X Y (Ω,P)
|(X, Y )| ≤ pV(X) V(Y)
n
X1, X2, X3
X1
X1X2X1+X2
X1X2

λ∈RZ=λX +YV(Z)≥0
V(Z) = λ2V(X)+2λCov(X, Y ) + V(Y)
V(X) = 0 (X, Y ) = 0 V(Z)
λ∈R
V(X)6= 0 ∆ = 4 (X, Y )2−4V(X) V(Y)≤0
V(Z)λ∈R
|Cov(X, Y )| ≤ pV(X) V(Y)
n p = 1/3
X1
n
p X1
n p = 1/3
V(X1) = np(1 −p) = 2n/9 V(X2)=2n/9X1, X2, X3
X1+X2=n−X3V(X1+X2) = V(n−X3) = V(X3)=2n/9
V(X1+X2) = V(X1) + 2 (X1, X2) + V(X2)
(X1, X2) = −n/9