Formulae Calculus

Formulae

• Addition rule: P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
P (A∩B)
P (B)

• Conditional Probability: P (A|B) =

provided P (B) > 0.

• Independence: A and B are independent if P (A ∩ B) = P (A) × P (B).
• Discrete random variable: µx =

P
x

• Continuous random variable: µx =

xp(x), σx2 =

R∞
−∞

P
x

x2 p(x) − µ2x .

xf (x)dx, σx2 =

R∞
−∞

x2 f (x) − µ2x .

• If X1 , . . . , Xn are random variables, the mean of any linear combination is given by
µc1 X1 +···+cn Xn = c1 µX1 + · · · + cn µXn .
• If X1 , . . . , Xn are independent random variables, the variance of any linear combination is given by
2
2
σc21 X1 +···+cn Xn = c21 σX
+ · · · + c2n σX
.
n
1

• Binomial distribution: p(x) =
special case with n = 1.

n
x




px (1 − p)n−x ; x = 0, 1, . . . , n. µx = np, σx2 = np(1 − p). Bernoulli is a

• Poisson distribution: p(x) = e−λ λx /x!; x = 0, 1, . . .. µx = λ, σx2 = λ.
• Hypergeometric: p(x) =

N −R
(R
x )( n−x )
, µx =
N
(n)

nR
,
N

σx2 =

nR
(1
N

−

R N −n
)
.
N N −1

• Geometric: p(x) = (1 − p)x−1 p; x = 1, 2, . . .. µx = p1 , σx2 =
• Negative binomial: p(x) =

x−1
r−1




1−p
.
p2

pr (1 − p)x−r ; x = r, r + 1, . . .. µx = pr , σx2 =

• Normal distribution: If X ∼ N (µ, σ 2 ), then Z =

X−µ
σ

r(1−p)
.
p2

∼ N (0, 1).

• Lognormal distribution: If Y ∼ LN (µ, σ 2 ), then X = loge (Y ) ∼ N (µ, σ 2 ). µy = eµ+σ
2
e2µ+σ .
• Exponential: f (x) = λe−λx , x > 0. µx =
• Uniform distribution: f (x) =

1
,
b−a

1
,
λ

σx2 =

2

/2

2

, σy2 = e2µ+2σ −

1
.
λ2

a < x < b. µx =

a+b
,
2

σx2 =

(b−a)2
.
12

• Central Limit Theorem: If X1 , . . . , Xn are independent random variables each with mean µ and standard
deviation σ, then the following hold approximately:
S = X1 + · · · + Xn ∼ N (nµ, nσ 2 )
X=

X1 + · · · + Xn
σ2
∼ N (µ,
)
n
n

provided n is large (n > 30).
• Large sample confidence interval for µ:

σ
X ± zα/2 √ .
n

Sample size needed to get a desired confidence bound B: n =

2
zα/2
σ2

B2

.

• Confidence interval for p:

r
p˜ ± zα/2
where n
˜ = n + 4, p˜ =

p˜(1 − p˜)
n
˜

X+2
.
n+4

Sample size needed for a desired confidence bound B: n =

1

2
zα/2
p∗ (1−p∗ )

B2

where p∗ is a guess of p.

• Small sample confidence interval for µ:
X ± tn−1,

s
.
n

α/2 √

• Large sample CI for µX − µY based on independent samples:

r
X − Y ± zα/2

2
σ2
σX
+ Y.
nX
nY

• CI for pX − pY :

r
p˜X − p˜Y ± zα/2
where p˜X =

X+1
,
nX +2

p˜Y =

Y +1
,
nY +2

p˜X (1 − p˜X )
p˜Y (1 − p˜Y )
+
n
˜X
n
˜Y

n
˜ X = nX + 2, n
˜ Y = nY + 2.

2
• Small-sample CI for µX − µY based on independent samples, when σX
6= σY2 :

r
X − Y ± tν,

α/2

s2X
s2
+ Y
nX
nY

where


ν=

s2
X
nX

+

/nX )2
(s2
X
nX −1

+

s2
Y
nY

2

/nY )2
(s2
Y
nY −1

,

(1)

rounded down to the nearest integer.
2
= σY2 :
• Small-sample CI for µX − µY based on independent samples, when σX

r
X − Y ± tnX +nY −2,
where

r
sp =

α/2 sp

1
1
+
nX
nY

(nX − 1)s2X + (nY − 1)s2Y
.
nX + nY − 2

• CI for paired data:
D ± tn−1,

sD
α/2 √

n

where D = X − Y .
• Large sample test for µ: Test statistic z∗ =

X−µ
√0.
σ/ n

(
P − value =

• Test for p: Test statistic z∗ = √

p−p
ˆ
0

if H1 : µ > µ0
if H1 : µ < µ0
if H1 : µ 6= µ0 .

(assume np0 > 10, n(1 − p0 ) > 10.)

p0 (1−p0 )/n

(
P − value =

• Small sample test for µ: Test statistic t∗ =

(
P − value =

P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)

P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)

if H1 : p > p0
if H1 : p < p0
if H1 : p 6= p0 .

X−µ
√0.
s/ n

P (tn−1 > t∗ )
P (tn−1 < t∗ )
2 × P (tn−1 > |t∗ |)

• Large sample test for µX − µY : Test statistic: z∗ = p

if H1 : µ > µ0
if H1 : µ < µ0
if H1 : µ 6= µ0 .

X−Y −∆0

2 /n +σ 2 /n
σX
X
Y
Y

(
P − value =

P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)

2

.

if H1 : µX − µY > ∆0
if H1 : µX − µY < ∆0
if H1 : µX − µY =
6 ∆0 .

(2)

p
ˆX −p
ˆY

• Large sample test for pX − pY : Test statistic: z∗ = √

p(1−
ˆ
p)(1/n
ˆ
X +1/nY )

pˆ =

X+Y
nX +nY

where pˆX =

X
nX

, pˆY =

Y
nY

and

.
P (Z > z∗ )
P (Z < z∗ )
2 × P (Z > |z∗ |)

(
P − value =

if H1 : pX > pY
if H1 : pX < pY
if H1 : pX 6= pY .

2
• Small-sample test for µX − µY , independent samples (assume σX
6= σY2 ): Test statistic: t∗ = p

P (tν > t∗ )
P (tν < t∗ )
2 × P (tν > |t∗ |)

(
P − value =

X−Y −∆0
s2
/nX +s2
/nY
X
Y

.

if H1 : µX > µY
if H1 : µX < µY
if H1 : µX 6= µY

where ν is as in (1).
2
• Small-sample test for µX − µY , independent samples (assume σX
= σY2 ): Test statistic: t∗ =

√X−Y −∆0
sp

1/nX +1/nY

,

where sp is as in (2).

(
P − value =

• Test for paired data: Test statistic: t∗ =

P (tnX +nY −2 > t∗ )
P (tnX +nY −2 < t∗ )
2 × P (tnX +nY −2 > |t∗ |)
D−µ
√0 ,
sD / n

(
P − value =

if H1 : µX > µY
if H1 : µX < µY
if H1 : µX 6= µY .

where D = X − Y .

P (tn−1 > t∗ )
P (tn−1 < t∗ )
2 × P (tn−1 > |t∗ |)

if H1 : µD > µ0
if H1 : µD < µ0
if H1 : µD 6= µ0 .

• Chi-Square test of goodness-of-fit: H0 : p1 = p10 , . . . , pk = pk0 . Test statistic:
χ2∗ =

k
X
(Oi − Ei )2

Ei

i=1

where Ei = N pi0 . P -value = P (χ2k−1 > χ2∗ ).
• Chi-Square test of homogeneity: H0 : p1j = p2j = · · · = pIj for each j (j = 1, . . . , J). Test statistic:
χ2∗ =

J
I
X
X
(Oij − Eij )2
i=1 j=1

where Eij =

Oi. O.j
O..

Eij

. P -value= P (χ2(I−1)(J−1) > χ2∗ ).

• Correlation between X and Y :

Pn
i=1

r = pPn

x2
i=1 i

xi yi − nxy

− nx2

pPn
i=1

yi2 − ny 2

• Test for ρ = ρ0 (where ρ0 6= 0): Use the test statistic W = 21 loge
1+ρ
2
1
distributed with mean µW = 21 loge 1−ρ
and variance σW
= n−3
.

.

1+r
,
1−r

which is approximately normally

√

r n−2
To test for ρ = 0, use U = √
, which has a tn−2 distribution under H0 .
2
1−r

Pn

xi yi −nxy
• Least squares regression coefficients: βˆ1 = Pi=1
, βˆ0 = y − βˆ1 x.
n
2
2
i=1

xi −nx

• 100(1 − α)% CI for β0 and β1 are: βˆ0 ± tn−2,α/2 sβˆ0 and βˆ1 ± tn−2,α/2 sβˆ1 where

s
sβˆ0 = s

1
x2
+ Pn
,
n
(xi − x)2
i=1

sβˆ1 = pPn
i=1

3

s
(xi − x)2

r
and

s=

(1 − r2 )

Pn

(yi − y)2
.
n−2
i=1

• 100(1 − α)% CI for the mean predicted value at x is βˆ0 + βˆ1 x ± tn−2,α/2 syˆ where

s
syˆ = s

(x − x)2
1
+ Pn
n
(xi − x)2
i=1

• 100(1 − α)% prediction interval at x is given by βˆ0 + βˆ1 x ± tn−2,α/2 spred where

s
spred = s

1+

(x − x)2
1
+ Pn
n
(xi − x)2
i=1

Pn

• Regression SS (SSR) = i=1 (ˆ
yi − y)2 , Error SS (SSE) =
Analysis of variance identity: SST = SSR + SSE.
• s2 =

SSE
.
n−p−1

Coefficient of determination R2 =

• To test H0 : β1 = · · · = βp = 0, use F =

Pn
i=1

(yi − yˆi )2 and Total SS (SST) =

SSR
.
SST

SSR/p
.
SSE/(n−p−1)

Under H0 , F ∼ Fp,n−p−1 .

• F -test for one-way ANOVA:
SST r =

I
X

2

2

Ji X i. − N X ..

i=1

SSE =

Ji
I
X
X

I
X

2
Xij
−

i=1 j=1

2

Ji X i.

i=1

SST r
SSE
M ST r
M ST r =
, M SE =
, F =
I −1
N −I
M SE
Under H0 : µ1 = · · · = µI , F ∼ FI−1, N −I .
• 100(1 − α)% CI for µi is X i. ± tN −I,

q
α/2

M SE
Ji

• Fisher’s Least Significant Difference Method:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± tN −I,α/2

q

M SE( J1i +

1
Jj

)

– To test H0 : µi = µj , reject H0 at level α if

r
|X i. − X j. | > tN −I,α/2

M SE(

1
1
+ )
Ji
Jj

• Bonferroni Method with C simultaneous comparisons:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± tN −I,α/(2C)

q

M SE( J1i +

1
Jj

)

– To test H0 : µi = µj , reject H0 at level α if

r
|X i. − X j. | > tN −I,α/(2C)

M SE(

1
1
+ )
Ji
Jj

• Tukey-Kramer Method for all possible comparisons:
– 100(1 − α)% CI for (µi − µj ) is X i. − X j. ± qI,N −I,α

q

M SE 1
( Ji
2

+

1
Jj

)

– To test H0 : µi = µj , reject H0 at level α if

r
|X i. − X j. | > qI,N −I,α

4

M SE 1
1
( + )
2
Ji
Jj

Pn
i=1

(yi − y)2 .