Understanding Expected Values and Covariance in Probability and Statistics
Classified in Other subjects
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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