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Formula Sheet for Final Exam
Summary Statistics
n ∑
i=
x i
n
s
2
n
i=
(xi − x¯)
2
n − 1
- Sample standard deviation:=
s
2
- Inter-quartile range = q 75 − q 25 , where qx = x
th
percentile.
Probability
c ) = 1 − P (A)
P (A or B) = P (A) + P (B) − P (A and B)
P (A|B) =
P (A and B)
P (B)
- If A and B are mutually exclusive:
P (A and B) = 0
- If A and B are independent:
P (A and B) = P (A)P (B)
P (A|B) =
P (B|A)P (A)
P (B)
,... , A
n
are mutually exclu-
sive and
n
i=
P (A
i
) = 1, then
P (B) =
n ∑
i=
P (B|Ai)P (Ai)
Discrete Distribution
• E(X) =
x∈X
xP (x)
• V (X) =
x∈X
(x − μ)
2 P (x) =
x∈X
x
2 P (x) − μ
2
p(x) =
p if x=
1 − p if x=
E(X) = p V (X) = p(1 − p)
- If X ∼ Binomial (n, p), for r = 0, 1 ,... n,
P (X = r) =
n
r
p
r (1 − p)
n−r
E(X) = np V (X) = np(1 − p)
n
r
n!
r!∗(n−r)!
where n! = n(n − 1)(n − 2) · · · 1
and 0! = 1
- If X ∼ Poisson ( λ ), for k = 0, 1 , 2 ,...
P (X = k) =
λ
k
k!
e
−λ
E(X) = λ, V (X) = λ
Continuous Distribution
) be the cumulative distribution func-
tion of X with density p(x):
F (xo) = P (X ≤ xo) =
xo
−∞
p(x)dx
P (a < X ≤ b) = F (b) − F (a)
- If X is Normal with mean μ and variance σ
2 :
X ∼ N (μ, σ
2 ), then
X − μ
σ
= Z ∼ N (0, 1)
Expectation and Variance
Let X and Y denote two independent random vari-
ables and let a and b denote two known constants.
2 V (X)
• V (X) = E(X
2 ) − [E(X)]
2
• E(X + Y ) = E(X) + E(Y )
• V (X + Y ) = V (X) + V (Y )
Sampling Distribution
Let X 1 ,... , Xn be an iid sample with E(Xi) = μ and
V (Xi) = σ
2
. Denote
X the sample mean and s
2 the
sample variance.
X ∼ N
μ,
σ
2
n
X − μ
σ/
n
∼ N(0, 1)
X − μ
s/
n
∼ t n− 1
Point Estimation
Let X and Y be estimators for θ.
- Bias = E(X) − θ
- MSE (X) = E[(X − θ)
2 ] = V(X) + [Bias(X)]
2
- Efficiency of X compared to Y =
M SE(Y )
M SE(X)
*MSE = mean squared error
Confidence Interval for One Mean
- If the population variance σ
2 is known:
¯x ± z (1−C%)/ 2
×
σ
n
where z (1−C%)/ 2
is the (1 − C%)/2 quantile of
the standard Normal distribution.
- If the population variance σ
2 is unknown:
x¯ ± t (1−C%)/ 2 , n− 1
×
s
√
n
Confidence Interval for the Difference
in Two Means
2
1
and σ
2
2
are known:
(x 1 − x 2 ) ± z (1−C%)/ 2
×
σ
2
1
n 1
σ
2
2
n 2
2
1
and σ
2
2
are unknown and equal:
(x 1
− x 2
) ± t (1−C%)/ 2 , n 1 +n 2 − 2
× s p
n 1
n 2
sp =
(n 1 − 1)s
2
1
2
2
n 1
2
1
and σ
2
2
are unknown and unequal:
(x 1 − x 2 ) ± t (1−C%)/ 2 , tWS
×
s
2
1
n 1
s
2
2
n 2
tWS =
(s
2
1
/n 1 + s
2
2
/n 2 )
2
(s
2
1
/n 1 )
2 /(n 1 − 1) + (s
2
2
/n 2 )
2 /(n 2 − 1)
Hypothesis Testing
- Type I error (α) = P (reject H 0 |H 0 is true)
- Type II error (β) = P (not reject H 0 |H 0 is false)
- Power = 1 − β
- Test statistic for one mean, H 0
: μ = μ 0
X − μ 0
σ ¯ X
- Test statistic for difference of two sample means,
H
0
: μ 1
− μ 2
= d 0
(X 1 − X 2 ) − d 0
σ X 1 −X 2
Joint Distribution
Let X and Y denote two random variables with joint
distribution p(x, y). Let a and b denote two known
constants
xy
g(x, y)p(x, y)
xy
(g(x, y) − E[g(x, y)])
2
p(x, y)
Independence: for K × R cells,
K ∑
i=
R ∑
j=
(O
ij
− E
ij
2
Eij
∼ χ
2
(K−1)(R−1)
ANOVA
Random samples of size ni from K Normal popula-
tions with equal variance:
M S
between
K
i=
ni(
Xi −
X)
2
K − 1
, where
X =
K
i=
ni
Xi
K
M S
within
K
i=
ni
j=
(Xij −
Xi)
2
K
i=
(ni − 1)
M S
between
M S
within
∼ F
K− 1 ,
∑
(ni−1)
