Binomial Distribution

Binomial Distribution:

X ~ b(n,p) 

"The distribution describes the probability of exactly r successes in N trials if the probability of a success in a single trial is p (we sometimes also use q = 1 − p, the probability for a failure, for convenience). It was first presented by Jacques Bernoulli in a work which was posthumously published" (1) 

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

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

LINK TO GOOGLE SPREADSHEET - CLICK HERE!

H	I	J	K	L	M	N	O
1	Binomial Distribution
2	X~b(n,p)	x	p	n	nCx	Answer	ROUND (4th)
3	Formula	8	0.45	9	9
4	P(X=x) = (nCx)*p^(x)*(1-p)^(n-x) =	⇒	⇒	⇒	⇒	0.008323487068359	0.0083
5	For x= (0, 1,2,3....n): p=% of success, n=total # of outcomes (samples). x=# of successes
6	Google Sheets Formula =					(=K3*(I3^H3)*(1-I3)^(J3-H3))
7	Urn problem WITH replacement
8	Google Sheets Function (Basic)
9	[=BINOMDIST(x,n,p,FALSE]
10	(more google sheets functions - see below)
11
12
13
14	Mean
15	E[X]=μ = np
16	μ =	⇒	⇒	⇒	⇒	4.05	4.05
17	Google Sheets Formula =					(=J3*I3)
18
19	Variance
20	Var = σ^(2)
21	σ^(2) = np(1-p)
22	σ^(2)=					2.2275	2.2275
23	Google Sheets Formula =					(=L16*(1-I3))
24
25
26	Standard Deviation
27	St.Dev = √σ	⇒	⇒	⇒	⇒	1.49248115565993	1.4925
28	Google Sheets Formula =					(=SQRT(L22))
29
30
31	Moment Generating Function
32	M(t) = E[e^(tX)] =
33	[(e^(t))*p+(1-p)]^(n)
34
35	E[X] from MGF
36	E[X]= ∂{[e^(t)p+(1-p)]^(n)}/∂x |0 =
37	np	⇒	⇒	⇒	⇒	4.05	4.05
38	Google Sheets Formula =					(=L3*K3)
39
40	E[X^(2)] from MGF
41	E[X]= ∂^(2){[e^(t)p+(1-p)]^(n)}/∂x^(2) |0 =
42	n(n-1)p^(2)+np	⇒	⇒	⇒	⇒	18.63	18.63
43	Google Sheets Formula =					(=L3*(L3-1)*K3^(2)+L3*K3)
44
45	E[X^(3)] from MGF
46	E[X]= ∂^(3){[e^(t)p+(1-p)]^(n)}/∂x^(3) |0 =
47	n(n-1)(n-2)p^(3)+3n(n-1)p^(2)+np	⇒	⇒	⇒	⇒	93.717	93.717
48	Google Sheets Formula =					(=L3*(L3-1)*(L3-2)*(K3^3)+3*L3*(L3-1)*(K3^2)+L3*K3)
49
50
51	Third Moment of X about the Mean =
52	E[(X-E[X])^(3)]=
53	np(1-p)*(1-2p)	⇒	⇒	⇒	⇒	0.22275	0.2228
54	Google Sheets Formula =					(=(L3*K3)*(1-K3)*(1-2*K3))
55
56
57	Distribution Function
58	0............................................t<0
59	(nCx)*(p^(x))*(1-p)^(n-x) .....0 ≤t ≤ n
60	1..........................................n ≤ t
61	(t is integer portion of t)
62
63	Other Moments/Kurtosis/Etc:
64	http://mathworld.wolfram.com/BinomialDistribution.html
65
66	To Plot Binomial Distribution:
67	To Plot Binomial Distrubtion use this link:
68	http://www.wolframalpha.com/input/?i=binomial+distribution+%2810%2C+.50%29
69
70	Google Sheets Function (All)
71	BINOMDIST(num_successes, num_trials, prob_success, cumulative)
72	P{X=x} [=BINOMDIST(x,n,p,0)]	⇒	⇒	⇒	⇒	0.008323487068359	0.0083
73	P{X≤x} [=BINOMDIST(x,n,p,1)]	⇒	⇒	⇒	⇒	0.999243319357422	0.9992
74	P{X<x} [=BINOMDIST(x-1,n,p,1)]	⇒	⇒	⇒	⇒	0.990919832289063	0.9909
75	P{X>x} [=1-BINOMDIST(x,n,p,1)]	⇒	⇒	⇒	⇒	0.000756680642578	0.0008
76	P{X≥x} [=1-BINOMDIST(x-1,n,p,1)]	⇒	⇒	⇒	⇒	0.009080167710937	0.0091
77
78	x<y
79	P{x<X<y} [=BINOMDIST(y-1,n,p,1)-[=BINOMDIST(x,n,p,1)]
80	P{x<X≤y} [=BINOMDIST(y,n,p,1)-[=BINOMDIST(x,n,p,1)]
81	P{x≤X<y} [=BINOMDIST(y-1,n,p,1)-[=BINOMDIST(x-1,n,p,1)]
82	P{x≤X≤y} [=BINOMDIST(y,n,p,1)-[=BINOMDIST(x-1,n,p,1)]
83
84
85
86	Definition
87	A Binomial Distribution is a Bernoulli event that happens many times.
88
89	A Binomial distribution also only has a “success/failure” outcome, but it is repeated many times.
90	•Examples:
91	–Probability of getting 7 heads in 40 coin tosses
92	–Probability of observing 14 boys of 20 babies
93	–Probability of 8 people getting a raise of 30 employees
94
95

LINK TO GOOGLE SPREADSHEET - CLICK HERE!