Basic Formulae
Pn
1
Mean: X̄ = n i=1 Xi
Pn
1
2
Sample Variance: s2 = n−1
i=1 (Xi − X̄) =
P
n
1
2
2
i=1 Xi − nX̄
n−1
Pn
1
Covariance: Cov(X, Y ) = n−1
i=1 (Xi −
Pn
1
X̄)(Yi − Ȳ ) = n−1 ( i=1 Xi Yi − nX̄ Ȳ )
)
Correlation: r = Cov(X,Y
sX sY
Ch 3: Discrete Random Variables
P
pmf Properties: p(x) ≥ 0, all x p(x) = 1
CDF: F (x) = P (XP
≤ x)
n
Finite GP Sum: ni=1 ri−1 = 1−r
, for r ̸= 1
P∞ i−1 1−r1
r = 1−r , for |r| < 1
Infinite GP Sum: i=1P
n−1
= (1 − r)−2 ,
1st Deriv Infinite GP: ∞
n=1 nr
for |r| < 1
P
n−2
=
2nd Deriv Infinite GP: ∞
n=2 n(n − 1)r
−3
2(1 − r) , for |r| < 1
P
Expectation (Mean): µ = E(X) = all x xp(x)
Expectation
of Function:
E[u(X)] =
P
all x u(x)p(x)
P
Variance: V ar(X) = E[(X − µ)2 ] = all x (x −
µ)2 p(x)
p
V ar(X) = E(X 2 )−[E(X)]2 , σ(X) = V ar(X)
Mean & Var Properties: E(a) = a, E(aX +
b) = aE(X) + b
V ar(a) = 0, V ar(aX + b) = a2 V ar(X), σ(aX +
b) = |a|σ(X)
Bernoulli: p(x) = π x (1 − π)1−x (x = 0, 1).
E(X) = π, V ar(X) = π(1
x − π) n−x
n
Binomial: p(x) = x π (1 − π)
(x = 0..n).
E(X) = nπ, V ar(X) = nπ(1P
− π) Binomial Exp: (a + b)n = nx=0 nx ax bn−x
Geometric: p(x) = π(1 − π)x−1 (x ≥ 1). F (x) =
1 − (1 − π)x . E(X) = π1 , V ar(X) = 1−π
Pπ∞2 λx
e−λ λx
λ
Poisson: p(x) = x! (x ≥ 0).
x=0 x! = e .
E(X) = V ar(X) = λ
Ch 4: Continuous Random Variables pX (x)pY (y), f (x, y) = fX (x)fY (y)
Rb
pdf Properties:
P
(a
≤
X
≤
b)
=
f (x)dx
a
R∞
f (x) ≥ 0, −∞ f (x)dx = 1
Rx
CDF: F (x) = P (X ≤ x) = −∞ f (t)dt. f (x) =
d
F (x). P (X = a) = 0 R
dx
∞
Expectation: E(X) = −∞ xf (x)dx
R∞
E[u(X)] = −∞ u(x)f (x)dx
R∞
Variance: V ar(X) = −∞ (x − µ)2 f (x)dx
1
(a < x < b). F (x) = x−a
Uniform: f (x) = b−a
b−a
(b−a)2
E(X) = a+b
, V ar(X) = 12
2
Exponential: f (x) = 1θ e−x/θ (x > 0). F (x) =
1 − e−x/θ (x ≥ 0). E(X) = θ, V ar(X) = θ2
Memoryless: P (X > s + t | X > s) = P (X > t)
2
1
Normal: f (x) = √2πσ exp − (x−µ)
. E(X) = µ,
2σ 2
Conditional: P (A|B) =
P (AB)
,
P (B)
pX|Y (x|y) =
p(x,y)
, fX|Y (x|y) = ffY(x,y)
pY (y)
(y)
Ch 6: Sampling Distribution
Sum Properties: E(X + Y ) = E(X) + E(Y ),
V ar(X + Y ) = V ar(X) + V ar(Y ) + 2Cov(X, Y )
2
Sample Mean Dist: E(X̄) = µ, V ar(X̄) = σn
X̄−µ
√ = √X̄−µ
∼ N (0, 1)
1-Sample Z: Z = σ/
n
V ar(X̄)
Diff btwn Means: E(X̄1 − X̄2 ) = µ1 − µ2 ,
σ2
σ2
V ar(X̄1 − X̄2 ) = n11 + n22
2-Sample Z: Z = (X̄√1 −2X̄2 )−(µ21 −µ2 ) ∼ N (0, 1)
σ1 /n1 +σ2 /n2
CLT (Prop): Z = √ π̂−π
∼ N (0, 1)
π(1−π)/n
2-Sample
CLT
(Prop):
Z
V ar(X) = σ 2
(π̂1 −π̂2 )−(π1 −π2 )
√
∼ N (0, 1)
π1 (1−π1 )/n1 +π2 (1−π2 )/n2
Linear Trans.: Y = a + bX ∼ N (a + bµ, b2 σ 2 )
∼ N (0, 1). ϕ(z) = Sample Var Exp: E(s2 ) = σ 2
Std Normal: Z = X−µ
σz
R
2
√1 exp(− 1 z 2 ). Φ(z) =
ϕ(t)dt. X = µ + Zσ Chi-Square Stat: χ2 = (n−1)s
∼ χ2n−1
2
−∞
2π
σ2
p
X̄−µ
√ ∼ tn−1 (T = Z/ Y /v)
t-Stat: T = s/
n
Ch 5: Joint Distributions
s2 /σ 2
=
1
1
Joint CDF/pmf: F (x, y) P
= PP
(X ≤ x, Y ≤ y), F-Stat: F = s22 /σ22 ∼ Fn1 −1,n2 −1
p(x, y) = P (X = x, Y = y), Px y p(x, y) = 1
Ch 7: Point & Interval Estimation
Marginal pmf: pX (x) =
y p(x, y), pY (y) =
P
Bias & MSE: Bias = E(θ̂) − θ. M SE =
x p(x, y)
Rx Ry
Joint pdf: F (x, y) = −∞ −∞ f (u, v)dvdu. E[(θ̂ − θ)2 ]
RR
CI Def: P [L(X) ≤ θ ≤ U (X)] = 1 − α
f (x, y)dxdy = 1
R
Z-Int Mean (Known σ): X̄ ± Zα/2 √σn (Length
RMarginal pdf: fX (x) = f (x, y)dy, fY (y) =
2Zα/2 √σn )
f (x, y)dx
Covariance: Cov(X, Y ) = E[(X − µX )(Y − Large Sample Z CI: X̄ ± Zα/2 √s
q n
µY )] = E(XY ) − µX µY
Z-Int Proportion: π̂ ± Zα/2 π̂(1−π̂)
n
Cov(aX +
b,
cY
+
d)
=
ac
·
Cov(X,
Y
)
RR
P P
t-Int
Mean
(Unknown
σ):
X̄
±
tα/2,n−1 √sn
E(XY): x y xy p(x, y) or
xy f (x, y)dxdy
(n−1)s2 (n−1)s2
Correlation: ρ = √ Cov(X,Y )
Chi-Sq CI Variance:
, χ2
V ar(X)V ar(Y )
χ2α/2
1−α/2
Independence: F (x, y) = FX (x)FY (y), p(x, y) =
2
h 2
i
1
s1 1
s21
1
or (Fα )−1
,
F-Int
Variances:
Sample Size Mean: n =
Fα
s22 Fα/2 s22 F1−α/2
Paired t-Int (Dep Pops): di = Xi − Yi .
Zα/2 2
√
Sample Size Prop: n = π(1 − π) E
(Max d¯ ± t √sd
(Z
approximation
Trap:
Don’t
divide
by
n twice!)
α/2 n
size: use π = 0.5)
Z-Test Means (Known σ): Z = √(X̄21 −X̄2 )−0
Test Assumptions
2
Zα/2 σ
E
σ1 /n1 +σ2 /n2
Ch 8: Hypothesis Testing
√0
Z-Test (Mean, Known σ): Z = X̄−µ
σ/ n
Z-Test Rejections: Two-sided: |Z| > Zα/2 .
One-sided Upper: Z > Zα . Lower: Z < −Zα
√0
t-Test (Mean, Unk σ): t = X̄−µ
s/ n
t-Test Rejections: Two-sided: |t| > tα/2 . Upper: t > tα . Lower: t < −tα
Z-Test Proportion: Z = √ π̂−π0
2
π0 (1−π0 )/n
2
= (n−1)s
σ02
2
Chi-Sq Test Variance: χ
Chi-Sq Rejections: Two-sided: χ > χ2α/2 or
< χ21−α/2 . Upper: χ2 > χ2α . Lower: χ2 < χ21−α
Type
II Error β (Two-Sided):
β =
[µ0 −µa ]
[µ0 −µa ]
P [σ/√n] − Zα/2 [≤]Z[≤] [σ/√n] + Zα/2
Type
II Error β
(Right-Tailed): β =
[µ0 −µa ]
P Z[≤] [σ/√n] [+Zα ]
Type
II Error β (Left-Tailed):
β =
[µ0 −µa ]
P Z[≥] [σ/√n] [−Zα ]
Ch 9: Inference for Two Populations
Z-Int
(Known σ): (X̄1 − X̄2 ) ±
q 2Means
σ1
σ22
Zα/2 n1 + n2
Z-Int
q Means (Large, Unk σ): (X̄1 − X̄2 ) ±
s2
s2
Zα/2 n11 + n22
Z-Int
Proportions:
q
Zα/2
(π̂1
−
π̂2 )
π̂1 (1−π̂1 )
2)
+ π̂2 (1−π̂
n1
n2
Pooled t-Int (Eq Var): s2p =
q
(X̄1 − X̄2 ) ± tα/2 sp n11 + n12
0): General: All require Random Samples (RS).
Z-Test/Int (Mean): Pop ∼ N OR n ≥ 30
π̄(1−π̄)(1/n1 +1/n2 )
(CLT). Known σ.
Z-Test Proportions (Unpooled, D0 ̸= 0):
t-Test/Int (Mean): Pop ∼ N (robust if n large).
1 −π̂2 )−D0
Z = r π̂ (π̂
Unknown σ.
1 (1−π̂1 ) + π̂2 (1−π̂2 )
n1
n2
Z-Test/Int (Prop): nπ ≥ 5 & n(1 − π) ≥ 5
X̄1 −X̄2 )−0
Pooled t-Test Means: t = (√
(Large n).
sp 1/n1 +1/n2
2 2
χ2 -Test/Int (Var): Pop STRICTLY ∼ N (not
F-Test Variances: F = s1 /s2
robust!).
Two-tailed: F > Fα/2 or F < F1−α/2
2-Samp Z/t (Unpooled): Indep RS. Both meet
Upper-tailed: F > Fα . Lower-tailed: F < F1−α
¯ d
d−µ
1-samp rules.
Paired t-Test: t = sd /√n
Pooled t-Test/Int: Indep RS, Pops ∼ N , Eq
Miscellaneous Equations
Vars (σ12 = σ22 ).
F-Test/Int (Vars): Indep RS, Pops STRICTLY
Approx. Distributions (Part
a)
approx
σ12
σ22
∼ N.
Means: X̄1 − X̄2 ∼ N µ1 − µ2 , n1 + n2
Paired t-Test/Int: Dep RS (pairs), Diffs di ∼ N .
approx
π2 (1−π2 )
1)
Props: π̂1 −π̂2 ∼ N π1 − π2 , π1 (1−π
+
n1
n2
Justifications / CLT (Part c)
approx
Mean CLT: If n ≥ 30 =⇒ X̄ ∼ N (µ, σ 2 /n)
approx
Prop CLT: If nπ ≥ 5, n(1 − π) ≥ 5 =⇒ π̂ ∼
N (π, π(1−π)
)
n
Independence: Indep. samples =⇒ V ar(X1 −
X2 ) = V ar(X1 ) + V ar(X2 )
Theoretical Properties
Chi-Square Addition: If X ∼ χ2ν1 , Y ∼ χ2ν2
(indep), then X + Y ∼ χ2ν1 +ν2
Definition of t-Ratio: If Z ∼ N (0, 1), Y ∼ χ2ν
(indep), then T = √Z ∼ tν
Z-Test Proportions (Pooled,
+n2 π̂2
π̂1 −π̂2
π̄ = n1nπ̂11 +n
. Z=√
2
±
D0
=
Y /ν
(n1 −1)s21 +(n2 −1)s22
n1 +n2 −2
Check if two-tailed! Check d.f.!
s2 ̸= V ar(X̄) Check p!
F-Distribution Inverse Rule:
F1−α
=