Formula Sheets for Math 441

7/27/2019 Formula Sheets for Math 441

1/9

Table 1: Special Discrete Distributions

Notation and Parameters Discrete pdf f (x) Mean Variance MGF M X (t)Binomial

X BIN (n, p ) nx px q n x np npq ( pet + q )n

0 < p < 1 x = 0 , 1, . . . , nq = 1 p

BernoulliX BIN (1, p) px q 1x p pq pet + q

0 < p < 1 x = 0 , 1q = 1 p

Negative Binomial

X NB (r, p ) x1r 1 pr q xr r/p rq/p 2 pe

t

1qe tr

0 < p < 1 x = r, r + 1 , . . .r = 1 , 2, . . .

GeometricX GEO ( p) pq x1 1/p q/p 2 pe

t

1qe t0 < p < 1 x = 1 , 2, . . .

q = 1 pHypergeometricX HY P (n,M,N ) M x

N M n x /N n nM/N n

M N 1 M N N nN 1 Not tractable

n = 1 , 2, . . . , N x = 0 , 1, . . . , nM = 0 , 1, . . . , N

PoissonX P OI () e

xx ! e

(e t 1)

0 < x = 0 , 1, . . .Discrete Uniform

X DU (N ) 1/N N +12N 2 112 1N e

t e ( N +1)t

1

e t

N = 1 , 2, . . . x = 1 , 2, . . . , N

1


2/9

Table 2: Special Continuous Distributions

Notation and Parameters Continuous pdf f (x) Mean Variance MGF M X (t)Uniform

X UNIF (a, b) 1baa + b

2(ba ) 212 e

bt eat(ba ) ta < b a < x < b

NormalX N (, 2) 1 2 e

[(x)/ ]2 / 2 2 et + 2 t 2 / 2

0 < 2

Gamma

X GAM (, ) 1 ( ) x 1ex/ 2 11t

0 < 0 < x0 <

ExponentialX EXP () 1

ex/ 2 1

1t0 < 0 < xTwo-Parameter Exponential

X EXP (, ) 1 e(x) / + 2 et

1t0 < < x

Double ExponentialX DE (, ) 12 e|x|/ 22 e

t

12 t 20 <

2


3/9

Normal distribution: X N (, 2)

f (x) =1

2 e1

2 2(x) 2 ;< x < , < < , > 0

Student t distribution: X t(v)

f (x) = v+12 v v2

1

1 + x 2v(v+1) / 2 ; < x < , v = 1 , 2, . . .

Chi-Square distribution: X 2(v)

f (x) =1

2v/ 2 v2xv/ 21ex/ 2 ; x > 0, v = 1 , 2, . . .

F distribution: X F (v1 , v2)

f (x) =

v1 + v22 v12 v22

v1v2

v1 / 2

xv1 / 2

1

; x > 0, v1 = 1 , 2, . . . , v2 = 1 , 2, . . .

3


4/9

Table 3: Summary of Condence Intervals (Single Population)

Parameter on Limits of CondenceNature of the Which Condence Procedure Interval with CondencePopulation Interval Is Set Coefficient 1 Quantitative data; , the population Draw a sample of size n and

standard deviation mean compute the value of x, x z/ 2 n is known; the estimate of population normal

Quantitative data; , the population Draw a sample of size n and

standard deviation mean compute x and x tn 1,/ 2 s n is not known;

population normal s = 1n 1 (x i x)2 tn 1,/ 2 is the value obtainedimportant when from the t distributionsample size is with n 1 degrees of small (n < 30) freedom.

Quantitative data; , the population Draw a sample of size n andstandard deviation mean compute x and s x z/ 2 s n is not known;

population not Condence interval isnecessarily normal approximatesample size islarge ( n 30)

Quantitative data; p, the probability Draw a sample of size n and

binomial case of success (the note x, the number of xn z/ 2 xn (1 xn )npopulation successes; obtain x/n , the Condence interval is basedproportion) estimate of p. Here on the central limit

n is assumed large. theorem, thus, isapproximate.

Quantitative data; 2 , the poplulation Draw a sample of size n and

population normal variance compute s2 (n 1) s2

X 2n 1 ,/ 2, (n 1) s

2

X 2n 1 , 1 / 2

4


5/9

Table 4: Summary of Tests of Hypotheses (Single Population)

Nature of the Null Alternative Test Decision Rule: Reject H 0

If Population Parameter Hypothesis Hypothesis Statistic the Computed Value Is

Quantitative data; = 0 1. > 0 1. greater than zvariance 2 is 2. < 0 X 0/ n 2. less than zknown; 3. = 0 3. less than z/ 2 orpopulation greater than z/ 2normallydistributed

Quantitative data; = 0 1. > 0 1. greater than tn 1,variance 2 is 2. < 0 X 0S/ n 2. less than tn 1,not known; 3. = 0 3. less than tn 1,/ 2 orpopulation greater than tn 1,/ 2normallydistributed;sample sizesmall (n < 30)

Quantitative data; = 0 1. > 0 1. greater than zvariance 2 is 2. < 0 X 0S/ n 2. less than znot known; 3. = 0 3. less than z/ 2 orpopulation not greater than z/ 2necessarily The level of signicance isnormal; sample approximately 100 large (n 30) percent.

Quantitative data; = 0 1. > 0 1. greater than 2n 1,population 2. < 0 (n 1) S

2

202. less than 2n 1,1

normally 3. = 0 3. less than 2n 1,1/ 2 ordistributed greater than 2n 1,/ 2

Quantitative data; p p = p0 1. p > p 0 1. greater than zbinomial case 2. p < p 0 (X/n ) p0 p 0 (1 p 0 )n

2. less than z3. p = p0 3. less than z/ 2 orHere n is greater than z/ 2

assumed The level of signicance islarge. approximately 100

percent.

5


6/9

Table 5: Summary of Condence Intervals (two populations)

Parameter on Limits of CondenceNature of the Which Condence Procedure Interval with Condence

Population Interval Is Set Coefficient 1 Quantitative data; 1 2 , the Draw samples of sizes m and n

variance 21 difference of from the two populations, get (x y) z/ 2 21m + 22nand 22 are the population the respective means x and y,known; both means and nd the value of x y,populations are an estimate of 1 2 .normallydistributed.

Quantitative data; 1 2 , the Draw samples of sizes m and n21 and 22 are difference of from the two populations; (x y) tm + n 2,/ 2s p 1m + 1nnot known but the population compute x y andassumed equal; means

both s p = (m 1) s 21 +( n 1) s 22m + n 2populations arenormallydistributed.

Quantitative data; 1 2 , the Draw samples of sizes m and n;21 and 22 are not difference of the compute x y; compute s21 x y z/ 2

s 21m +

s 22n

known and not population means and s22 . Condence interval is approximate.assumed equal;populations maynot be normal;sample sizes mand n are large.

Quantitative data; p1 p2 , the Draw two samples, one from p1 is the difference of the each population, of sizes m xm yn proportion in population and n; nd x/m and y/n ,

Population 1 and proportions the estimates of p1 and p2 z/ 2 xm (1 xm )m +

yn (1 yn )n p2 in Population 2; respectively; then getsample sizes m x/m y/n . Condence interval is based on theand n are assumed central limit theorem, thus islarge. approximate.

6


7/9

Table 6: Summary of Tests of Hypotheses (Two Populations)

Nature of the Null Alternative Test Decision Rule: Reject H 0Population Parameter Hypothesis Hypothesis Statistic If the Computed Value Is

Quantitative data; 1 2 1 = 2 1. 1 > 2 1. greater than zvariance 21 and 2. 1 < 2 2. less than z22 are known; 3. 1 = 2 3. less than z/ 2 orboth populations X Y

21m + 22ngreater than z/ 2

normallydistributed.

Quantitative data; 1 2 1 = 2 1. 1 > 2 1. greater than z21 , 22 unknown 2. 1 < 2 2. less than zpopulations not 3. 1 = 2 3. less than

z/ 2 or

necessarily X Y S 21m + S 22n

greater than z/ 2

normal; samplesizes are large.

Quantitative data; 1 2 1 = 2 1. 1 > 2 1. greater than tm + n 2,21 and 22 are not 2. 1 < 2 X Y S p 1m + 1n 2. less than tm + n 2,known but are 3. 1 = 2 3. less thanassumed to be where tm + n 2,/ 2 orequal; both S p = (m 1) S 21 +( n 1) S 22m + n 2 greater thanpopulations are tm + n 2,/ 2normallydistributed.

Paired observations 1 2 D = 0 1. D > 0 1. greater than tn 1,( = 0) i.e., 2. D < 0 d

ns d 2. less than tn 1,

1 = 2 3. D = 0 where 3. less than tn 1,/ 2d = di /n or greater than

and tn 1,/ 2sd = 1n 1 (di d)2

Quantitative data; p1 p2 p1 = p2 1. p1 > p 2 1. greater than zbinomial case 2. p1 < p 2

Xm

Y n

p(1 p)( 1m + 1n ) 2. less than z3. p1 = p2 where 3. less than z/ 2 or

p = X + Y m + n greater than z/ 2Both m and n are The level of signicance is

assumed large. approximately100 percent.

7


8/9

T a b

l e

7 : S u m m a r y o f t h e F o r m u l a s

f o r I n f e r e n c e s a b o u t a M e a n

( ) , o r a D i ff e r e n c e o f t w o M e a n s

( 1 2

)

C o n

d e n c e i n t e r v a l = P o i n t e s t i m a t o r

( T a b

l e d v a

l u e )

( E s t i m a t e d o r t r u e s t

d . d e v .

)

T e s t s t a t i s t i c =

P o i n t e s t i m a t o r

P a r a m e t e r v a

l u e a t H

0

( n u

l l h y p o t h e s i s )

( E s t i m a t e d o r t r u e ) s t

d .

d e v o

f p o i n t e s t i m a t o r

S i n g

l e S a m p l e

I n d e p e n

d e n t

S a m p l e s

M a t c h e d P a i r s

P o p u l a t i o n ( s )

N o r m a l w i t h

N o r m a l

N o r m a l

N o r m a l

f o r t h e

G e n e r a l

u n k n o w n

1 = 1 =

1 = 2

G e n e r a l

d i ff e r e n c e

D i =

X i

Y i

I n f e r e n c e o n

M e a n

M e a n

1 2 =

1

2 =

1 2 =

D

S a m p l e ( s )

X 1 , . . . , X n

X 1 , . . . , X n

X 1 , . . . , X n 1

X 1 , . . . ,

X n 1

X 1 , . . . , X n 1

D 1 =

X 1

Y 1

. . .

Y 1 , . . . , Y n 2

Y 1 , . . . ,

Y n 2

Y 1 , . . . , Y n 2

D n

= X

n

Y n

S a m p l e s i z e n

L a r g e

n 2

n 1

2

n 1

2

n 1

3 0

n 2

n 3 0

n 2

2

n 2

2

n 2

3 0

P o i n t e s t i m a t o r

X

X

X

Y

X

Y

X

Y

D

= X

Y

V a r i a n c e o f P o i n t e s t i m

a t o r

2 n

2 n

2

1 n

1

+ 1 n

2

2 1 n 1 +

2 2 n 2

2 1 n 1

+ 2 2 n

2

2 D n

E s t i m a t e d s t

d . d e v

.

S n

S n

S P

1 n

1

+ 1 n

2

S 2 1

n 1

+ S

2 2 n 2

S

2 1 n 1

+ S

2 2 n 2

S D

n

D i s t r i b u t i o n

N o r m a l

t w i t h

t w i t h

t w i t h

N o r m a l

t w i t h

d . f

. = n

1

d . f

. = n 1

+ n 2

2

d . f

. e s t i m a t e d

d . f

. = n

1

T e s t s t a t i s t i c

X 0

S / n

X 0

S / n

( X

Y ) 0

S P

1 n

1

+ 1 n

2

( X

Y ) 0

S

2 1 n 1

+ S

2 2 n 2

( X

Y ) 0

S

2 1 n 1

+ S

2 2 n 2

D D

, 0

S D

/ n

S D =

s a m p l e s t

d .

S 2 P =

( n 1

1 ) S

2 1

+ ( n

2

1 ) S

2 2

n 1

+ n 2

2

d e v .

o f t h e

D i s

d . f

. =

[ ( s 2 1 n

1

) + ( s

2 2 n 2

) ] 2 / [ ( n 1

1 )

1 ( s

2 1 n 1

) 2 + ( n

2

1 )

1 ( s

2 2 n 2 )

2 ]

8


9/9

Table 8: Condence intervals, tests of hypotheses, and prediction intervals for straight-line regression analysis.

DistributionParameter 100(1 )% Condence Interval H 0 Test Statistic( T ) Under H 0 1 1 tn 2,1/ 2S 1 1 =

(0)1 T =

( 1

(0)

1)

S 1 tn 2 0 0 tn 2,1/ 2S 0 0 =

(0)0 T =

( 0 (0)0 )

S 0tn 2

Y |X 0 Y + 1(X 0 X ) tn 2,1/ 2S Y X 0 Y |X 0 = (0)Y |X 0 T =

Y + 1 (X 0 X )(0)Y | X 0

S Y X 0tn 2

Y X 0 Y + 1(X 0 X )tn 2,1/ 2S Y |X 1 + 1n + (X 0 X ) 2(n 1) S 2X

Note: Y |X = 0 + 1X is the assumed true regression model. 1 =

n1 (X i X )( Y i Y )n

1 (X i X ) 2 0 = Y 1X Y =

0 +

1X =

Y +

1(X

X )tn 2,1/ 2 is the 100(1 / 2)% point of the t distribution with n 2 degrees of freedom.

S 2Y |X =1

n 2n1 (Y i Y i )2 S 1 =

S Y | XS X n 1

S 2Y =1

n 1n1 (Y i Y )2 S 0 = S Y |X 1n + X 2(n 1) S 2X

S 2X =1

n 1n1 (X i X )2 S Y X 0 = S Y |X 1n + (X 0 X ) 2(n 1) S 2X

9

Formula Sheets for Math 441

Documents

Transcript of Formula Sheets for Math 441