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Statistical Formulas Cheat Sheet for Final Exam, Cheat Sheet of Statistics

This summary cheat sheet on Statistics is a good review to prepare for the final exam

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Formula Sheet for Final Exam
Summary Statistics
Sample mean: ¯x=
n
P
i=1
xi
n
Sample variance:
s2=Pn
i=1(xi¯x)2
n1
Sample standard deviation:= s2
Inter-quartile range = q75 q25, where qx=xth
percentile.
Probability
Complement: P(Ac) = 1 P(A)
Addition law:
P(Aor B) = P(A) + P(B)P(Aand B)
Conditional probability:
P(A|B) = P(Aand B)
P(B)
If A and B are mutually exclusive:
P(Aand B)=0
If A and B are independent:
P(Aand B) = P(A)P(B)
Bayes’ rule:
P(A|B) = P(B|A)P(A)
P(B)
Partition law: If A1, . . . , Anare mutually exclu-
sive and Pn
i=1 P(Ai) = 1, then
P(B) =
n
X
i=1
P(B|Ai)P(Ai)
Discrete Distribution
E(X) = P
xX
xP (x)
V(X) = P
xX
(xµ)2P(x) = P
xX
x2P(x)µ2
If XBernoulli (p),
p(x) = (pif x=1
1pif x=0
E(X) = p V (X) = p(1 p)
If XBinomial (n, p), for r= 0,1,...n,
P(X=r) = n
rpr(1 p)nr
E(X) = np V (X) = np(1 p)
n
r=n!
r!(nr)! where n! = n(n1)(n2) ···1
and 0! = 1
If XPoisson ( λ), for k= 0,1,2, . . .
P(X=k) = λk
k!eλ
E(X) = λ, V (X) = λ
Continuous Distribution
Let F(xo) be the cumulative distribution func-
tion of Xwith density p(x):
F(xo) = P(Xxo) = Zxo
−∞
p(x)dx
P(a<Xb) = F(b)F(a)
If Xis Normal with mean µand variance σ2:
XN (µ, σ2), then
Xµ
σ=ZN (0,1)
Expectation and Variance
Let Xand Ydenote two independent random vari-
ables and let aand bdenote two known constants.
E(aX +b) = aE(X) + b
1
pf3
pf4
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Formula Sheet for Final Exam

Summary Statistics

  • Sample mean: ¯x =

n ∑

i=

x i

n

  • Sample variance:

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

  • Complement: P (A

c ) = 1 − P (A)

  • Addition law:

P (A or B) = P (A) + P (B) − P (A and B)

  • Conditional probability:

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)

  • Bayes’ rule:

P (A|B) =

P (B|A)P (A)

P (B)

  • Partition law: If A 1

,... , 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

  • If X ∼ Bernoulli (p),

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

  • Let F (x o

) 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.

  • E(aX + b) = aE(X) + b
  • V (aX + b) = a

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

  • If σ

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

  • If σ

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

  • (n 2 − 1)s

2

2

n 1

  • n 2
  • If σ

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

  • E[g(X, Y )] =

xy

g(x, y)p(x, y)

  • V [g(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)