Final Formula Sheet for STAT 2800

Discrete Distribution: Mean: )(xpxx

Variance: )()( 22 xpxx

Continuous Distribution:

Mean:

dxxfxx )(

Variance:

dxxfxx )()( 22 or

222 )( dxxfxx

Median: 5.0)(~

dxxf

Binomial Distribution: xnx

xnx

nxp

)1(

)!(!

!)( , for nx ,...,1,0

where

x

n

xnx

n

)!(!

!

Mean: nx

Variance: )1(2 nx

Poisson Distribution:

( )!

xep x

x

, with 0 for nx ,..3,2,1,0

Mean: x Variance: x

2 Normal Distribution:

)2()( 22

2

1)(

xexf , for x

Mean: x

Variance: 22 x

Standard Normal: 2)( 2

2

1)( zezf

, where

x

z

Exponential Distribution: xexf )( , with 0 for 0x

cecxp )(

Mean:

1x

Variance: 2

2 1

x

Weibull Distribution: )(1)( xexxf , for 0x

tetxp 1)( Mean:

1

11

x

Variance:

2

22 1

12

11

x

Lognormal Distribution:

)2/(])[ln( 22

2

1)(

xe

xxf , for 0x

where

xz

ln

Mean: 2

2

)(

eYE Variance: 22 222)( eeYV Sample Mean:

n

xxxx n

...21

Sample Variance:

1

22

2

n

n

xx

s

ii

Interquartile Range: 13 QQIQR

Mild Outliers: )(5.11 IQRQ , )(5.13 IQRQ

Extreme Outliers: )(31 IQRQ , )(33 IQRQ

Disjoint:

( ) ( )P A B P A P B

0P A B

Non-Disjoint Addition Rule: ( ) ( )P A B P A P B P A B

Non-Disjoint Multiplication Rule: )()|()( BPBAPBAP

Independent Events: )()()( BPAPBAP

| ( )P A B P A

| ( )P B A P B

Conditional Probability:

|P A B

P A BP B

|P A B

P B AP A

Complement: )(1)()( APAPAP

Central Limit Theorem:

x xn

xz

n

Bound on Error: n

szB critical (rearrange

for sample size)

Final Formula Sheet for STAT 2800

Tests concerning a single mean: x

z

n

or

nsx

t

with df 1 n

Single Mean Confidence Intervals:

n

szx critical or

n

stx critical

Tests concerning a difference between two means: Independent Data:

2

22

1

21

2121 )(

nn

xxz

or

2

22

1

21

2121 )(

n

s

n

s

xxt

with

1

)(

1

)(

])()[(

2

42

1

41

222

21

n

se

n

se

sesedf

and Confidence Intervals:

2

22

1

21

21 )(nn

zxx critical

or

2

22

1

21

21 )(n

s

n

stxx critical

Pooled Data:

21

2121

11

)(

nns

xxt

p

with df 221 nn , 2

)1()1(

21

222

211

nn

snsns p

and Confidence Interval:

2121

11)(

nnstxx pcritical

Paired Data:

ns

dt d with

df 1 n , n

dd

n

ii

1 , 1

1

2

12

2

nn

d

ds

n

i

n

ii

i

and Confidence Interval: n

std critical

Tests Concerning :

w

wWZ

or 21

2

r

nrU

with df 2 n and

Confidence Interval: wzW 2

where r

rW

1

1ln

2

1 ,

1

1ln

2

1W

, 3

12

nW

and

1

12

2

w

w

e

e

Pearson’s Sample Correlation Coefficient:

yyxx

xy

SS

S

yVarxVar

yxCovr

)()(

),(

n

xxS i

ixx

22 ,

n

yyS i

iyy

22 ,

n

yxyxS ii

iixy

Least Squares Regression Line: xy 10

ˆˆˆ with

xx

xy

xx

yy

S

S

S

Sr

xVar

yVarr

)(

)(1

and xy 10

Residuals: iii yye ˆ

Coefficient of Determination:

SST

SSEr 12 where yySSST and

xyyy SSSSE 1̂

Quality Control:

The R Chart:

k

iiR

kR

1

1 with

RDUCL 4 and RDLCL 3 and

2

^

d

R

The x Chart:

k

i

ixk

x1

1 with

RAxUCL 2 and RAxLCL 2 or

sAxUCL 3 and sAxLCL 3

The s Chart:

k

iis

ks

1

1 with

sBUCL 4 and sBLCL 3 and 4

^

c

s

The p Chart:

k

iip

kp

1

ˆ1 with

n

pppUCL

)1(3

and

n

pppLCL

)1(3

The c Chart:

k

iic

kc

1

1 with

ccUCL 3 and ccLCL 3 Average run length:

pARL

1