Geometric Distributions

Below is my Google Docs Spreadsheet for Geometric Distributions:

This sheet you cannot change because I have it locked, but if you would like access so you can use my sheet, and just put in your data and get the answer - email me: messenger1964@yahoo.com

	Geometric Distribution
		x	p	Answer
	Formula	0	0.25
	p(x) = p(1-p)^(x) =	-->	-->	0.25
	where p is the probability of success, 0<p<1 and x = {0,1,2.....}
	Google Sheets Function
	[=NEGBINOMDIST(x,1,p)]	-->	-->	0.25
	(more google sheets functions - see below)
	Mean
	E[X]=μ = (1-p)/p
	μ =	-->	-->	3
	Variance
	Var = σ^(2)
	σ^(2) = (1-p)/p^(2)
	σ^(2)=	-->	-->	12
	Standard Deviation
	St.Dev = √σ	-->	-->	3.46410161513775
	Third Moment of X about the Mean =
	E[(X-E[X])^(3)]=
	(1-p)*(2-p)/p^(3)=	-->	-->	84
	4th Moment of X about the Mean =
	E[(X-E[X])^(4)]=
	(1-p)*(p^(2)-9p+9)/p^(4)=	-->	-->	1308
	Moment Generating Function
	M(t) = E[e^(tX)] = p*(1/((1-e^(t))*(1-p))
	p*(1/((1-e^(t))*(1-p))
	Distribution Function
	∀t < 0 ......F(t) = P{X≤t} =P{X∉R} = 0
	∀t ≥ 0 ......F(t) = P{X≤[t]} = 1-(1-p)^([t]+1)
	Google Sheets Function (All)
	BINOMDIST(num_successes, num_trials, prob_success, cumulative)
	P{X=x} [=NEGBINOMDIST(x,1,p)]
	P{X≤x} [=NEGBINOMDIST(x,1,p)]
	P{X<x} [=NEGBINOMDIST(x,1,p)]
	P{X>x} [=1-NEGBINOMDIST(x,1,p)]
	P{X≥x} [=1-NEGBINOMDIST(x-1,1,p)]
	x<y
	P{x<X<y} [=NEGBINOMDIST(y-1,1,p)-[=NEGBINOMDIST(x,1,p)]
	P{x<X≤y} [=NEGBINOMDIST(y,1,p)-[=NEGBINOMDIST(x,1,p)]
	P{x≤X<y} [=NEGBINOMDIST(y-1,1,p)-[=NEGBINOMDIST(x-1,1,p)]
	P{x≤X≤y} [=NEGBINOMDIST(y,1,p)-[=NEGBINOMDIST(x-1,1,p)]
	http://mathworld.wolfram.com/GeometricDistribution.html
	http://www.math.uah.edu/stat/bernoulli/Geometric.html
	http://en.wikipedia.org/wiki/Geometric_distribution#Moments_and_cumulants

Hypergeometric Distributions

X ~Hyp(N,n,p) 

"This distribution describes the experiment where elements are picked at random without
replacement. More precisely, suppose that we have N elements out of which M has a certain
attribute (and N −M has not). If we pick n elements at random without replacement p(r)
is the probability that exactly r of the selected elements come from the group with the
attribute." (1)

(1)  Hand-book on STATISTICAL DISTRIBUTIONS for Experimentalists, by Christian Walck, Published by: Particle Physics Group Fysikum, University of Stockholm

Q

R

S

T

U

V

W

X

Y

Z

AA

AB

AC

1

Hypergeometric Distribution

2

a

d

x

n

N

NCn

dCx

N-dCn-x

Answer

ROUND (4th)

3

Formula

10

4

8

60

2558620845

210

230300

4

P(X=x) = {(dCx)*([N-d]C[n-x])}/(NCn) =

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.018901979984455

0.0189

5

For x= (0, 1...min(n.d)),

6

Google Sheets Formula =

(=(Y3*Z3)/X3)

(=ROUND(AA,4))

7

8

b

d

y

n

N

NCn

dCx

N-dCn-x

Answer

ROUND (4th)

9

15

3

8

60

2558620845

455

1221759

10

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.217265620299439

0.2173

11

12

13

14

15

16

Google Sheets Function (Basic)

17

[=HYPGEOMDIST(x,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.018901979984455

0.0189

18

(=HYPGEOMDIST(U4,V4,T4,W4))

19

Google Sheets Formula =

20

Mean

21

E[X]=μ = n*(d/N)

22

μ =

⇒

⇒

⇒

⇒

⇒

⇒

⇒

1.33333333333333

23

Google Sheets Formula =

(=V3*(T3/W3))

24

25

Variance

26

Var = σ^(2)=

27

n*(d/N)*(1-(d/N)*((N-n)/(N-1))

28

σ^(2)=

⇒

⇒

⇒

⇒

⇒

⇒

⇒

1.13747645951036

1.1375

29

Google Sheets Formula =

(=Q3*(O3/R3)*(1-(O3/R3)*((R3-Q3)/(R3-1))))

30

31

32

Standard Deviation

33

St.Dev = √σ

⇒

⇒

⇒

⇒

⇒

⇒

⇒

1.06652541437622

1.0665

34

Google Sheets Formula =

(=SQRT(V19))

35

36

37

Moment Generating Function

38

A moment generating function does exist for the hypergeometric distribution

39

40

41

42

Third Moment of X about the Mean =

43

E[(X-μ)^(3)]=

44

n*(d/N)*(1-d/N)*(1-(2d/n))*((N-n)/(N-1))*((N-2n)/(N-2))=

⇒

⇒

⇒

⇒

⇒

⇒

⇒

-1.11435807519969

-1.1144

45

Google Sheets Formula =

(=R3*(P3/S3)*(1-(P3/S3))*(1-(2*P3/R3))*((S3-R3)/(S3-1))*((S3-2*R3)/(S3-2)))

46

47

48

Google Sheets Function (All)

49

HYPGEOMDIST(num_successes, num_draws, successes_in_pop, pop_size)

50

P{X=x} [=HYPGEOMDIST(x,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.018901979984455

51

P{X≤x} [=HYPGEOMDIST(x,n,d,N)] Note: no cumulative in this function in Google Sheets.

0.018901979984455

52

P{X<x} [=HYPGEOMDIST(x-1,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.099370409061136

53

P{X>x} [=1-HYPGEOMDIST(x,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.018901979984455

54

P{X≥x} [=1-HYPGEOMDIST(x-1,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.900629590938864

55

56

x<y

57

P{x<X<y} [=HYPGEOMDIST(y-1,n,d,N)] - [=HYPGEOMDIST(x,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.31535282047622

58

P{x<X≤y} [=HYPGEOMDIST(y,n,d,N)] - [=HYPGEOMDIST(x,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.198363640314984

59

P{x≤X<y} [=HYPGEOMDIST(y-1,n,d,N)] - [=HYPGEOMDIST(x-1,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.234884391399539

60

P{x≤X≤y} [=HYPGEOMDIST(y,n,d,N)] - [=HYPGEOMDIST(x-1,n,d,N)]

⇒

⇒

⇒

⇒

⇒

⇒

⇒

0.117895211238303

61

62

63

LINK TO GOOGLE SPREADSHEET - CLICK HERE!