# Formulas Exam 2-1

```E(Y ) =
Pn
i=1 yi P r(Y
= yi )
σY2 = E(Y 2 ) − [E(Y )]2
p
σY = σY2
=y)
P r(Y = y|X = x) = P r(X=x,Y
P r(X=x)
P
P r(Y |X = x) = ni=1 P r(Y = yi |X = x)
P P
cov(X, Y ) = ni=1 m
j=1 (xi −&micro;X )(yj −&micro;Y )P r(X = xi , Y = yj )
ρXY = corr(X, Y ) = √
cov(X,Y )
var(X)var(Y )
=
σXY
σX σY
Pn
Ȳ =
1
n
s2y =
1
n−1
i=1 Yi
Pn
i=1 (Yi
− Ȳ )2
q
√
SE(Ȳ ) = s2y /n = sy / n
tact =
Ȳ Actual −&micro;Y
SE(Ȳ )
tact =
(Ȳm −Ȳw )−d)
SE(Ȳm −Ȳw )
or tact =
p − value = 2Φ(−|tact |)
q 2
SE(Ȳm − Ȳw ) = nsww +
βˆ1 −β1,0
SE(βˆ1 )
s2m
nm
95%C.I. = [βˆ1 − 1.96SE(βˆ1 ), βˆ1 + 1.96SE(βˆ1 )]
\Y ) = sxy = 1 Pn (Xi − X̄)(Yi − Ȳ )
cov(X,
i=1
n−1
\Y ) = rxy =
corr(X,
βˆ1 =
sxy
sx sy
Pn
(X −X̄)(Yi −Ȳ )
i=1
Pn i
2
i=1 (Xi −X̄)
=
sxy
s2x
s
= rxy sxy
βˆ0 = Ȳ − βˆ1 X̄
1
ESS
T SS
R2 =
=1−
SSR
T SS
R2 = (rxy )2
ESS =
Pn
− Ȳ )2
T SS =
Pn
− Ȳ )2 with T SS = ESS + SSR
SSR =
Pn
− Ŷi )2 =
i=1 (Ŷi
i=1 (Yi
i=1 (Yi
R̄2 = 1 −
n−1 SSR
n−k−1 T SS
SER = sû =
s2û =
1
n−k−1
σβ21 =
q
Pn
=1−
2
i=1 (ûi )
n−1
n−k−1 (1
SSR
n−k−1
ˆ2
i=1 ui
Pn
=
SSR
n−k−1
1 var[(Xi −&micro;x )ui ]
n [var(Xi )]2
F =
2
2
(RU
nrestricted −RRestricted )/q
2
(1−RU nrestricted )/(n−kU nrestricted −1)
F =
(SSRRestricted −SSRU nrestricted )/q
(SSRU nrestricted )/(n−kU nrestricted −1)
F = 1q {
t21 +t22 −2ρ̂t1 ,t2 t1 t2
}
1−ρ̂2t1 ,t2
∼ Fq=2,n−k−1
2
− R2 ) = 1 −
s2u
s2Y
```