91影视

The standard deviation is a measurement of a collection of values' variance or dispersion. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation suggests that the values are dispersed over a larger range.

a)

The Covariance between \({\rm{X}}\)and \({\rm{Y}}\) is

\({\rm{Cov(X,Y) = E((X - E(X))(Y - E(Y)))}}\)

The following holds by the definition of covariance

\(\begin{aligned}{\rm{Cov(X,Y + Z) &= E(X(Y + Z)) - E(X) \\times E(Y + Z) \\ & = E(XY) + E(XZ) - E(X)E(Y) - E(X)E(Z) \\ &= E(XY) - E(X)E(Y) + E(XZ) - E(X)E(Z) \\ &= Cov(X,Y) + Cov(X,Z) \end{aligned}\)

Therefore, \({\rm{Cov(X,Y + Z) = Cov(X,Y) + Cov(X,Z)}}\).

(b):

The following holds from\({\rm{(a)}}\)(twice)

\(\begin{array}{l}{\rm{Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{Y}}_{\rm{1}}}{\rm{ + }}{{\rm{Y}}_{\rm{2}}}} \right)\\{\rm{ = Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{1}}}} \right){\rm{ + Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right){\rm{ + Cov}}\left( {{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{Y}}_{\rm{1}}}} \right){\rm{ + Cov}}\left( {{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)\\{\rm{ = 5 + 1 + 2 + 8}}\\{\rm{ = 16}}\end{array}\)

Therefore, the covariance between the two total scores is \(16.\)