Understanding Expected Values and Covariance in Probability and Statistics

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1) E(???????? + ????) = ∑ (???????????? + ????)????(???? = ????????)     2)

????????????

(????) = ????[(???? − ????͓)² ]

= ∑ ????????????????(???? = ????????) + ∑ ????????(???? = ????????)              =????[????²+(????͓)²− 2????͓ ????]

= ????∑ ????????????(???? = ????????) + ???? ∑ ????(???? = ????????)

=????[????]²+????͓²− 2????͓ ????[????]

= ????????(????) + ????

=????[????]²−????͓²

= ???????????? + c

3) ????????????(???????? + ????) = ????{[(???????? + ????) − (a????͓ + ????)]²}     6) ????????????(????, ????)

= ????{[????(???? − ????͓)] ²}

=????[(????– ????͓)(???? − ????͓)]

=????² ????[(???? − ????????) ² ]

= ????????????(????)

= ????²  ????????????(????) ≡ ????²  ????͓

≡ ????͓²

4) ????(???????? + ???????? + ????) = ∑k∑m (???????????? + ???????????? + ????)????(???? = ???????? ∩ ???? = ????????)

= = ∑i∑j ????????????????(???????????????? , ????????) + ∑i ∑j ????????????????(???????????????? , ????????) + ∑i ∑j ????????(????????, ????????)

= ???? ∑k ????????∑m ????(???????? , ????????) + ????∑m yj∑k ????(???????? , ????????) + c

= ???? ∑k ????????????(???? = ???????? ) + ???? ∑m ????????????(???? = ????????) + c

= ????????[????] + ????????[????] + ???? =  ????????͓ + ???????????? + ????

5) ????????????(????, ????) ????[(???? − ????͓)(???? − ???????????? )]     7) ????????????(????????, ???? + ????????)

= ????(???????? − ???????????? − ????????͓ + ????͓ ???????? )

= ????[(????????– ????????͓)(???? + ???????? − ???? − ???????????? )]

= ????(????????) – ????????????͓ − ????͓ ???????? + ????͓ ????????                 = ????[????????(???? − ????͓)(???? − ???????? )]

= ????(????????)−????͓ ????????

= ????????????[(???? − ????͓)(???? − ???????? )]

= ????????????????????(????, ????) ≡ ???????? = ????͓y

8) = ????[{(???????? + ????????) − (???????? ͓ + ???????????? )}{???? − ???????? }]     10) ????(???????? + ???????? + ????|????)

= ????[{????(???? − ????????) + ????(???? − ???????? )}{???? − ???????? }]               =????????(????|????) + ????????(????|????) + ????

= ????????[(???? − ????????)(???? − ???????? )] + ????????[(???? − ???????? )(???? − ???????? )]

= ????????????????(????, ????) + ????????????????(????, ????)  ≡ ????????͓???? + ????????Yz

9) ????????????(∑???? ???????????????? + ????)

= ∑s ????????²????????????(???????? ) + 2 ∑s ∑????≠????????????????????????????????(???????? , ???????? )

≡ ∑s ????????² ????????????????² + 2∑???? ∑????≠????????????????????????X????Xt