Bernoulli Distributions:

What is a Bernoulli Distribution?

X~Bernoulli(p) 

"The Bernoulli distribution, named after the swiss mathematician Jacques Bernoulli (1654–1705), describes a probabilistic experiment where a trial has two possible outcomes, a success or a failure.  The parameter p is the probability for a success in a single trial, the probability for a failure thus being 1 − p (often denoted by q). Both p and q is limited to the interval from zero to one." (1)

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

Below is my Google Docs Spreadsheet for Bernoulli Distributions:

 Please email me if you would like a copy of the spreadsheet or what more information.   (info@internationalmathematics.org)

(Note: Bernoulli Distribution is on page: Math Stat Distributions I)

LINK TO GOOGLE SPREADSHEET - CLICK HERE!

A	B	C	D	E	F
1	Bernoulli Distribution
2	X~Bernoulli(p)	x	p	Answer	ROUND (4th)
3	Formula Probability Mass Function:	0	0.1
4	P(X=x) = p^(x)*(1-p)^(1-x) =	⇒	⇒	0.9	0.9
5	For x= (0,1)
6	Google Sheets Formula =			(=(D3^C3)*(1-D3)^1-C3)
7	Google Sheets Function
8	n/a
9
10
11	Mean
12	E[X]=μ = p
13	μ =	⇒	⇒	0.1	0.1
14	Google Sheets Formula =			(=D3)
15
16	Variance
17	Var = σ^(2)
18	σ^(2) = p(1-p)
19	σ^(2)=	⇒	⇒	0.09	0.09
20	Google Sheets Formula =			(=D3*(1-D3))
21
22
23	Standard Deviation
24	St.Dev = √σ	⇒	⇒	0.3	0.3
25	Google Sheets Formula =			(=SQRT(E19))
26
27
28	Third Moment of X about the Mean =
29	E[(X-E[X])^(3)]=
30	p(1-p)*(1-2p)=	⇒	⇒	0.072	0.072
31	Google Sheets Formula =			(=D3*(1-D3)*(1-2*D3))
32
33	Skew
34	If E[(X-E[X])^(3)] < 0 then skews to left.
35	If E[(X-E[X])^(3)] = 0 then symmetric
36	If E[(X-E[X])^(3)] > 0 then skews to right.
37	Skew=	⇒	⇒	Skews To Right
38	Google Sheets Formula =			(=IF(E30<0,"Skews To Left","Skews To Right"))
39
40	4th Moment of X about the Mean =
41	E[(X-E[X])^(4)]=
42	p(1-p)*(1-3p+3p^(2))=	⇒	⇒	0.0657	0.0657
43	Google Sheets Formula =			(=D3*(1-D3)*(1-3*D3+3*D3^2))
44
45
46
47	5th Moment of X about the Mean =
48	E[(X-E[X])^(5)]=
49	p(1-p)*(1-2p) (1-2p+2p^(2))	⇒	⇒	0.05904	0.059
50	Google Sheets Formula =			(=D3*(1-D3)*(1-2*D3)*(1-2*D3+2*D3^2))
51	Moment Generating Function
52	M(t) = E[e^(tX)] = 1-p+pe^(t)
53	1-p+pe^(t)
54
55	E[X] from MGF
56	E[X]= ∂{1-p+pe^(t)}/∂x |0 = p	⇒	⇒	0.1
57
58	E[X^(2)] from MGF
59	E[X^(2)]= ∂^(2){1-p+pe^(t)}/∂x^(2) |0 = p	⇒	⇒	0.1
60
61	E[X^(k)] from MGF
62	E[X^(k)]= ∂^(k){1-p+pe^(t)}/∂x^(k) |0 = p	⇒	⇒	0.1
63
64
65
66
67
68
69	Definition
70	The Bernoulli distribution, named after the Swiss mathematician Jacques Bernoulli (1654–1705),
71	This distribution describes a probabilistic experiment where a trial has two possible outcomes, a success or a failure.
72
73	Success =	p
74	Failure =	1-p or 'q'
75	Limits =	Both p and q are limited to range of 0 to 1
76	Other =	All Bernoulli are Independent trials
77
78
79	Examples:
80	–Probability of a head in a single coin flip
81	–Probability of having a boy
	–Probability of getting a raise

LINK TO GOOGLE SPREADSHEET - CLICK HERE!