Mathematical Statistics 214
Complete Exam Cheat Sheet
Descriptive Statistics
Probability Distributions
Ch 8: Estimation & CIs
Multivariate
Ch 10: Hypothesis Testing
01 Descriptive Statistics
1.1 Measures of Location
Arithmetic Mean
Median & Quantiles
Median = middle value (odd n) or average of two middle values (even
n
ȳ =
n).
1
∑ yi
n i=1
Q p : position = p(n + 1) (interpolate if not integer)
Weighted Mean
Sturges' Rule
ȳ w =
∑ wi yi
∑ wi
k = 1 + 3.322 log 10 (n),
Geometric Mean
Sample Mean from Frequency Table
G = (∏ y i )
n
w=
1/n
= antilog(
i=1
1
∑ log y i )
n
ȳ =
Harmonic Mean
∑ fi mi
∑ fi
where m i = class midpoint, f i = frequency.
H=
n
n
1
y
i=1 i
∑
Trimmed Mean (drop g values each end)
w
ȳ t =
1
∑ yi
n − 2g middle
Mean inequality always holds: H ≤ G ≤ ȳ arithmetic
=
( (xnj + 1)
-
2n .
max − min
k
1.2 Measures of Spread
Sample Variance
s2 =
IQR & Fences
∑ y 2i − nȳ 2
n−1
=
IQR = Q 3 − Q 1
∑(y i − ȳ) 2
n−1
Outlier fences: Q 1 − 1.5 ⋅ IQR,
Standard Deviation
Q 3 + 1.5 ⋅ IQR
Tchebyshe"'s Theorem (any distribution)
s = √s 2
P (|Y − µ| < kσ) ≥ 1 −
1
,
k2
k>1
Coe!cient of Variation
CV =
s
× 100%
ȳ
1.3 Shape: Skewness & Kurtosis
Skewness g 1
g1 =
Kurtosis g 2
y − ȳ
n
∑( i
)
s
(n − 1)(n − 2)
3
g2 =
g 1 > 0: right-skewed | g 1 < 0: left-skewed | g 1 = 0:
4
n(n + 1)
3(n − 1) 2
y − ȳ
∑( i
) −
s
(n − 1)(n − 2)(n − 3)
(n − 2)(n − 3)
g 2 > 0: leptokurtic (heavy tails) | g 2 < 0: platykurtic ("at)
symmetric
Right-skewed ⇒ mean > median > mode
02 Probability Distributions Reference
2.1 Discrete Distributions
Distribution
Binomial Y ∼ Bi(n, p)
Poisson Y ∼ Po(λ)
Geometric
Y ∼ Geom(p)
PMF p(y)
n
( )p y (1 − p) n−y
y
e −λ λ y
y!
p(1 − p) y−1
E(Y )
Var(Y )
When to use
np
np(1 − p)
Fixed n independent trials;
constant p; count successes
λ
λ
Count of rare events in a #xed
interval; mean rate λ
1
p
1−p
p2
Trials until first success
y−1 r
(
)p (1 − p) y−r
r−1
Neg.\ Binomial
NB(r, p)
Hypergeometric
Hyp(N, M, n)
M
N −M
( )(
)
y
n−y
N
( )
n
Multinomial
(n; p 1 , … , p k )
n!
p y1 ⋯ p ykk
y1 ! ⋯ yk ! 1
r
p
r(1 − p)
p2
nM
N
n⋅
E(Y i ) = np i
Trials until r-th success
(geometric is r = 1)
M N −M N −n
⋅
⋅
N
N
N −1
Sampling without replacement;
#nite pop N , M successes
np i (1 − p i );
Cov(Y i , Y j ) = −np i p j
n trials, k categories with #xed
probabilities
2.2 Continuous Distributions
Distribution
PDF f(y)
E(Y )
Var(Y )
When to use
Uniform
Y ∼ U(θ 1 , θ 2 )
$$\frac{1}{\theta_2-\theta_1}, \quad
\theta_1
θ1 + θ2
2
(θ 2 − θ 1 ) 2
12
Equal probability over
[θ 1 , θ 2 ]; CDF is linear
µ
σ2
Bell-shaped symmetric
data. Standardise:
Y −µ
Z=
σ
β
β2
Waiting time between
Poisson events;
memoryless; β = 1/λ
αβ
αβ 2
Sum of α independent Exp
(β) RVs; waiting for α-th
event
Sum of ν squared standard
normals; variance tests
Normal
Y ∼ N(µ, σ 2 )
(y − µ) 2
1
exp[−
]
2σ 2
σ√2π
Exponential
Y ∼ Exp(β)
1 −y/β
,
e
β
Gamma
Y ∼ Γ(α, β)
y>0
y α−1 e −y/β
,
β α Γ(α)
y>0
Chi-Squared
Y ∼ χ 2 (ν)
Special case: Γ( ν2 , 2)
ν
2ν
Beta
Y ∼ Beta(α, β)
$$\frac{y^{\alpha-1}(1-y)^{\beta-1}}
{B(\alpha,\beta)},\quad 0
α
α+β
(α + β) 2 (α + β + 1)
t-Distribution
T ∼ t(ν)
T=
Z
√χ 2 (ν)/ν
; symmetric bell, heavy
αβ
0
ν
ν−2
Small samples (n < 30), σ
unknown. As ν → ∞:
t → N(0, 1)
ν2
ν2 − 2
—
Ratio of two sample
variances; always positive
tails
F -Distribution
F ∼ F (ν 1 , ν 2 )
χ 2 (ν 1 )/ν 1
F=
χ 2 (ν 2 )/ν 2
Proportions/probabilities
in (0, 1)
2.3 Key Distribution Relationships
CLT
If Y 1 , … , Y n iid with mean µ, variance σ 2 :
Ȳ ≈ N(µ,
σ2 )
(n ≥ 30)
n
F from Variances
F=
S 12 /σ 21
∼ F (n 1 − 1, n 2 − 1)
S 22 /σ 22
Sample Var Pivot
F Reciprocal
(n − 1)S 2
∼ χ 2 (n − 1)
σ2
t-Pivot
1
∼ F (ν 2 , ν 1 )
F (ν 1 , ν 2 )
Normal Approx to Binomial
−µ
T = Ȳ
∼ t(n − 1)
S/√n
(if np ≥ 5,
n(1 − p) ≥ 5)
Y ≈ N(np, np(1 − p))
03 Multivariate Probability (Ch. 5)
3.1 Bivariate Distributions
Marginal Distributions
Covariance
Discrete: p 1 (y 1 ) = ∑ p(y 1 , y 2 )
Cov(Y 1 , Y 2 ) = E(Y 1 Y 2 ) − E(Y 1 ) E(Y 2 )
y2
Continuous: f 1 (y 1 ) = ∫
∞
−∞
E(Y 1 Y 2 ) = ∬ y 1 y 2 f(y 1 , y 2 ) dy 1 dy 2
f(y 1 , y 2 ) dy 2
Conditional Distribution
f(y 2 ∣ y 1 ) =
Independent ⇒ Cov = 0. But Cov = 0 does NOT imply
f(y 1 , y 2 )
f 1 (y 1 )
Independence Test
independence!
Correlation
ρ=
Cov(Y 1 , Y 2 )
,
σ1 σ2
−1 ≤ ρ ≤ 1
Y 1 ⊥ Y 2 ⟺ f(y 1 , y 2 ) = f 1 (y 1 ) ⋅ f 2 (y 2 ) ∀ (y 1 , y 2 )
Linear Combination (Theorem 5.12)
U = a1 Y1 + a2 Y2
E(U) = a 1 E(Y 1 ) + a 2 E(Y 2 )
Var(U) = a 21 Var(Y 1 ) + a 22 Var(Y 2 ) + 2a 1 a 2 Cov(Y 1 , Y 2 )
3.2 Bivariate Normal
f(y 1 , y 2 ) =
1
2πσ 1 σ 2 √1 − ρ 2
exp{−
2ρ(y 1 − µ 1 )(y 2 − µ 2 )
(y 1 − µ 1 ) 2
(y − µ 2 ) 2
1
−
+ 2
[
]}
2
σ1 σ2
2(1 − ρ 2 )
σ1
σ 22
σ
Conditional: Y 2 ∣ Y 1 = y 1 ∼ N(µ 2 + ρ 2 (y 1 − µ 1 ), σ 22 (1 − ρ 2 ))
σ1
KEY: For bivariate normal, Y 1 ⊥ Y 2 ⟺ ρ = 0
3.3 Conditional Expectation Theorems
Law of Total Expectation
Law of Total Variance
Var(Y 2 ) = Var[E(Y 2 ∣ Y 1 )] + E[Var(Y 2 ∣ Y 1 )]
E[E(Y 2 ∣ Y 1 )] = E(Y 2 )
04 Scenario → Distribution Decision Map
Scenario / Clue Words
Distribution
Test Statistic / Key Formula
Fixed n trials; success/fail; constant p
Binomial Bi(n, p)
Z=
p^ − p 0
√p 0 (1 − p 0 )/n
Trials until !rst success
Geometric Geom(p)
E(Y ) = 1/p
Trials until r-th success
Neg.\ Binomial NB(r, p)
E(Y ) = r/p
Count of rare events; rate λ; #xed interval
Poisson Po(λ)
P (Y = y) = e −λ λ y /y!
Sampling without replacement; #nite population
Hypergeometric
E(Y ) = nM/N
Equal probability on [θ 1 , θ 2 ]
Uniform U(θ 1 , θ 2 )
F (y) =
Bell-shaped; symmetric; heights/weights/errors
Normal N(µ, σ 2 )
Z=
Waiting time; memoryless; exponential decay
Exponential Exp(β)
E(Y ) = β = 1/λ
Sum of α exponentials; waiting for α-th event
Gamma Γ(α, β)
E(Y ) = αβ; Var(Y ) = αβ 2
Proportion/probability strictly in (0, 1)
Beta Beta(α, β)
E(Y ) =
n trials, k categories (not just 2)
Multinomial
Cov(Y i , Y j ) = −np i p j
Testing/CI for σ 2 ; single sample
χ 2 (n − 1)
C=
Ratio of two variances
F (n 1 − 1, n 2 − 1)
F = S 12 /S 22
Small sample (n < 30), σ unknown, testing/CI for µ
t(n − 1)
Large sample (n ≥ 30) OR σ known; CI/test for µ
Z ∼ N(0, 1)
T=
y − θ1
θ2 − θ1
Y −µ
∼ N(0, 1)
σ
α
α+β
(n − 1)S 2
σ 20
Ȳ − µ 0
S/√n
Z=
Ȳ − µ 0
σ/√n
05 Chapter 8 — Estimation & Confidence Intervals
All CIs use the form: Point estimate ± (critical value) × (standard error). Every CI here is two-sided unless labelled otherwise.
5.1 CI for Mean µ
Case A
Large sample (n ≥ 30) or σ known
Ȳ ± z α/2 ⋅
Case B
Small sample (n < 30) AND σ unknown
σ
√n
Ȳ ± t n−1, α/2 ⋅
Use S if σ unknown but n ≥ 30. Distribution: Z ∼ N(0, 1).
df = n − 1. Distribution: t(n − 1). Tail: two-sided.
Tail: two-sided.
Sample size for desired margin B:
n=(
z α/2 ⋅ σ
B
)
2
5.2 CI for Proportion π
p^ ± z α/2 √
p^(1 − p^)
n
Validity: n^
p ≥ 5 and n(1 − p^) ≥ 5. Distribution: Z ∼ N(0, 1). Tail: two-sided.
Sample size: n = z 2α/2 π(1 − π)/B 2 (use π = 0.5 if unknown)
5.3 CI for Variance σ 2
Pivot:
(n − 1)S 2
∼ χ 2 (n − 1)
σ2
(
(n − 1)S 2
,
χ 2n−1, α/2
(n − 1)S 2
)
χ 2n−1, 1−α/2
χ 2 is asymmetric — the lower and upper critical values are di$erent! Tail: two-sided.
5.4 CI for Ratio of Variances σ 21 /σ 22
S
√n
(
S 12
1
S2
F ν1 ,ν2 , α/2
⋅
2
,
S 12
S 22
⋅ F ν2 ,ν1 , α/2 )
ν 1 = n 1 − 1, ν 2 = n 2 − 1. Lower critical value: F lower = 1/F ν2 ,ν1 , α/2 (swap df!). Tail: two-sided.
5.5 CI for Difference in Means µ 1 − µ 2
Case
Condition
I: Large
n 1 , n 2 ≥ 30
II: Small, equal σ 2
σ 21 = σ 22 (verify with F -test)
Confidence Interval
(Ȳ 1 − Ȳ 2 ) ± z α/2 √
σ 21 ≠ σ 22 (Welch)
+
—
S 22
n2
n1 + n2 − 2
(n 1 − 1)S 12 + (n 2 − 1)S 22
n1 + n2 − 2
S 12
n1
+
(S 12 /n 1 ) 2
+
ν (Welch)
S 22
n2
(S 12 /n 1 + S 22 /n 2 ) 2
n1 + 1
Natural pairs; D i = Y 1i − Y 2i
n1
(Ȳ 1 − Ȳ 2 ) ± t ν, α/2 √
where ν =
IV: Paired
S 12
1
1
(Ȳ 1 − Ȳ 2 ) ± t n1 +n2 −2, α/2 ⋅ S p √
+
n1
n2
where S p2 =
III: Small, unequal σ 2
df
(S 22 /n 2 ) 2
−2
n2 + 1
D̄ ± t n−1, α/2 ⋅
SD
√n
n−1
5.6 CI for Difference in Proportions π 1 − π 2
(^
p 1 − p^2 ) ± z α/2 √
p^1 (1 − p^1 )
p^ (1 − p^2 )
+ 2
n1
n2
06 Chapter 10 — Hypothesis Testing
6.1 Framework (6 Steps)
6.2 Error Types
1. State H 0 (uses =) and H 1 (determines tail)
2. Choose α (typically 0.05)
3. Identify test statistic
4. State rejection region
H 0 TRUE
H 0 FALSE
Don't Reject
Correct (1 − α)
Type II Error (β)
Reject H 0
Type I Error (α)
Correct: Power (1 − β)
5. Calculate test statistic from data
Increasing n decreases both α and β simultaneously.
6. Conclude: Reject H 0 or Fail to Reject H 0
σ
NEVER say "Accept H 0 " — always "Fail to Reject H 0 "
6.3 Tail Direction Rule
UPPER TAIL
LOWER TAIL
TWO-SIDED
H1 : θ > θ0
H1 : θ < θ0
H1 : θ ≠ θ0
RR: test stat > c α
RR: test stat < c 1−α
RR: |test stat|> c α/2
All α in right tail
All α in left tail
α/2 in each tail
6.4 All Tests — Rejection Regions & CIs
Z and t are symmetric → use ± for two-sided. χ 2 and F are asymmetric → two-sided needs TWO di$erent critical values.
Parameter
H1
µ
Large n or σ
known
µ > µ0
Test Statistic
Ȳ − µ 0
Z=
σ/√n
upper
µ < µ0
Rejection Region
Distribution
Z > zα
Z ∼ N(0, 1)
Z < −z α
lower
|Z| > z α/2
µ ≠ µ 0 two
µ
Small n, σ
unknown
µ > µ0
T=
upper
Ȳ − µ 0
S/√n
µ < µ0
T > t n−1, α
t(n − 1)
T < −t n−1, α
lower
|T | > t n−1, α/2
µ ≠ µ 0 two
π
Proportion
π > π0
upper
Z=
p^ − π 0
Z > zα
Z ∼ N(0, 1)
√π 0 (1 − π 0 )/n
π < π0
Z < −z α
lower
|Z| > z α/2
π ≠ π 0 two
σ2
Variance
σ 2 > σ 20
upper
σ 2 < σ 20
lower
(n − 1)S 2
C=
σ 20
C > χ 2n−1, α
C < χ 2n−1, 1−α
χ 2 (n − 1)
C < χ 21−α/2 or C > χ 2α/2
σ 2 ≠ σ 20 two
σ 21 /σ 22
Variance Ratio
σ 21 > σ 22
F=
upper
F > F n1 −1,n2 −1, α
S 12
S 22
σ 21 < σ 22
F<
lower
µ1 − µ2 ≠ D0
µ1 − µ2
Small, equal σ 2
µ1 − µ2 ≠ D0
µD
Paired (
µ D ≠ 0 two
F > F n1 −1,n2 −1, α/2
Z=
two
T=
two
π 1 ≠ π 2 two
(Ȳ 1 − Ȳ 2 ) − D 0
Z=
|Z| > z α/2
Z ∼ N(0, 1)
|T | > t n1 +n2 −2, α/2
t(n 1 + n 2 − 2)
|T | > t n−1, α/2
t(n − 1)
|Z| > z α/2
Z ∼ N(0, 1)
√S 12 /n 1 + S 22 /n 2
(Ȳ 1 − Ȳ 2 ) − D 0
S p √1/n 1 + 1/n 2
D̄
T=
S D /√n
D i = Y 1i − Y 2i )
π1 − π2
Di" in
proportions
1
F n2 −1,n1 −1, α
F < F lower, α/2 or
σ 21 ≠ σ 22 two
µ1 − µ2
Large samples
F (n 1 − 1, n 2 − 1)
p^1 − p^2
√p^(1 − p^)(1/n 1 + 1/n 2 )
where p^ =
n 1 p^1 + n 2 p^2
n1 + n2
07 Quick Reference Card
Common Critical Values
Level
Common Mistakes
z α/2 (two-
z α (one-
sided)
sided)
E O 0s
1 64 S
=
.
.
90%
(α = 0.10)
1.6449
1.2816
-
.
.
01
Key Formulas
.
.
1.6449
=
2
.
32 C6
2.5758
2.0537
=
(asymmetric!)
Lower F critical: ALWAYS 1/F ν2 ,ν1 , α/2 — swap the df!
.
2.3263
z0 . 005
99%
(α = 0.01)
1 46
1.9600
E 0
98%
(α = 0.02)
Proportion test: use π 0 (not p^) in denominator of Z stat
χ 2 and F two-sided tests need two different critical values
EO Ors
95%
(α = 0.05)
Never "Accept H 0 " — always "Fail to Reject H 0 "
. 576 .
2
2.3263
Paired data → use T on di"erences D i , not a two-sample
test
Independence: must hold for ALL (y 1 , y 2 ), not just a few
points
Standard error of mean:
If a CI for a difference contains 0 → no significant difference
σ
√n
If a CI for a ratio contains 1 → no significant ratio difference
Pooled variance:
Right-skewed: mean > median > mode
S p2 =
(n 1 − 1)S 12 + (n 2 − 1)S 22
n1 + n2 − 2
g 2 = 0 is normal (mesokurtic); g 2 > 0 heavy tails; g 2 < 0 flat
Welch df:
ν=
S2
S2
( 1 + 2)
n1
n2
2
(S 12 /n 1 ) 2
(S 2 /n 2 ) 2
+ 2
n1 + 1
n2 + 1
−2
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