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1Bernoulli Distribution1Binomial Distribution1Hypergeometric Distribution
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2 X~Bernoulli(p)xpAnswerROUND (4th)2X~b(n,p)xpnnCxAnswerROUND (4th)2adxnNNCndCxN-dCn-xAnswerROUND (4th)Note:
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3Formula Probability Mass Function: 00.23Formula100.61530033Formula555201550411N= Total Sample (Largest #)
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4P(X=x) = p^(x)*(1-p)^(1-x) =⇒⇒0.80.84P(X=x) = (nCx)*p^(x)*(1-p)^(n-x) =⇒⇒⇒⇒0.18593784480.18594P(X=x) = {(dCx)*([N-d]C[n-x])}/(NCn) =⇒⇒⇒⇒⇒⇒⇒0.0000644994840.0001n= Sample space to Choose from Total (N)
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5For x= (0,1)5For x= (0, 1,2,3....n): p=% of success, n=total # of outcomes (samples). x=# of successes5For x= (0, 1...min(n.d)), x=No of success you want from sample space
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6Google Sheets Formula = (=(D3^C3)*(1-D3)^1-C3)6Google Sheets Formula = (=K3*(I3^H3)*(1-I3)^(J3-H3))6Google Sheets Formula = (=(Y3*Z3)/X3)(=ROUND(AA,4))
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7Google Sheets Function7Urn problem WITH replacement 7without replacementd=Sample space you are choosing desired result from.
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8n/a8Google Sheets Function (Basic) 8bdynNNCndCxN-dCn-xAnswerROUND (4th)SO.....
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99[=BINOMDIST(x,n,p,FALSE]915386025586208454551221759Combinations Formula
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1010(more google sheets functions - see below)10⇒⇒⇒⇒⇒⇒⇒0.21726562030.2173
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11Mean 11this is WITH replacement11(12) (13)..........(d)....(N-d)
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12E[X]=μ = p1212.(4) .(8)............(x)......(n-x)
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13μ =⇒⇒0.20.21313----------------------------------------------
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14Google Sheets Formula = (=D3)14Mean14.(25)........................(N)
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1515E[X]=μ = np15.(12).........................(n)
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16Variance16μ =⇒⇒⇒⇒9916Google Sheets Function (Basic)
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17Var = σ^(2)17Google Sheets Formula = (=J3*I3)17[=HYPGEOMDIST(x,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.0000644994840.0001IMPORTANT: note that 8+4 = 12 and 12+13 = 25.
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18σ^(2) = p(1-p) 1818(=HYPGEOMDIST(U4,V4,T4,W4))
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19σ^(2)=⇒⇒0.160.1619Variance19Google Sheets Formula =
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20Google Sheets Formula = (=D3*(1-D3))20Var = σ^(2)20Mean
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2121σ^(2) = np(1-p) 21E[X]=μ = n*(d/N)
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2222σ^(2)=3.63.622μ =⇒⇒⇒⇒⇒⇒⇒1.25
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23Standard Deviation23Google Sheets Formula = (=L16*(1-I3))23Google Sheets Formula = (=V3*(T3/W3))
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24St.Dev = √σ⇒⇒0.40.42424
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25Google Sheets Formula = (=SQRT(E19))2525Variance
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2626Standard Deviation26Var = σ^(2)=
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2727St.Dev = √σ⇒⇒⇒⇒1.8973665961.897427 n*(d/N)*(1-(d/N)*((N-n)/(N-1))
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28Third Moment of X about the Mean = 28Google Sheets Formula = (=SQRT(L22))28σ^(2)=⇒⇒⇒⇒⇒⇒⇒0.74013157890.7401
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29E[(X-E[X])^(3)]= 2929Google Sheets Formula = (=Q3*(O3/R3)*(1-(O3/R3)*((R3-Q3)/(R3-1))))
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30p(1-p)*(1-2p)=⇒⇒0.0960.0963030
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31Google Sheets Formula = (=D3*(1-D3)*(1-2*D3))31Moment Generating Function31
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3232M(t) = E[e^(tX)] = 32Standard Deviation
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33Skew33[(e^(t))*p+(1-p)]^(n) 33St.Dev = √σ⇒⇒⇒⇒⇒⇒⇒0.8603090020.8603
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34If E[(X-E[X])^(3)] < 0 then skews to left.3434Google Sheets Formula = (=SQRT(V19))
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35If E[(X-E[X])^(3)] = 0 then symmetric35E[X] from MGF35
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36If E[(X-E[X])^(3)] > 0 then skews to right.36E[X]= ∂{[e^(t)p+(1-p)]^(n)}/∂x |0 =36
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37Skew= ⇒⇒Skews To Right37np⇒⇒⇒⇒9937Moment Generating Function
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38Google Sheets Formula = (=IF(E30<0,"Skews To Left","Skews To Right"))38Google Sheets Formula = (=L3*K3)38A moment generating function does exist for the hypergeometric distribution
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404th Moment of X about the Mean = 40E[X^(2)] from MGF40
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41E[(X-E[X])^(4)]= 41E[X]= ∂^(2){[e^(t)p+(1-p)]^(n)}/∂x^(2) |0 =41
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42p(1-p)*(1-3p+3p^(2))=⇒⇒0.08320.083242n(n-1)p^(2)+np⇒⇒⇒⇒84.684.642Third Moment of X about the Mean =
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43Google Sheets Formula = (=D3*(1-D3)*(1-3*D3+3*D3^2))43Google Sheets Formula = (=L3*(L3-1)*K3^(2)+L3*K3)43E[(X-μ)^(3)]=
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444444n*(d/N)*(1-d/N)*(1-(2d/n))*((N-n)/(N-1))*((N-2n)/(N-2))=⇒⇒⇒⇒⇒⇒⇒-0.4111842105-0.4112
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4545E[X^(3)] from MGF45Google Sheets Formula = (=R3*(P3/S3)*(1-(P3/S3))*(1-(2*P3/R3))*((S3-R3)/(S3-1))*((S3-2*R3)/(S3-2)))
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4646E[X]= ∂^(3){[e^(t)p+(1-p)]^(n)}/∂x^(3) |0 =46
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475th Moment of X about the Mean = 47n(n-1)(n-2)p^(3)+3n(n-1)p^(2)+np⇒⇒⇒⇒825.48825.4847
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48E[(X-E[X])^(5)]= 48Google Sheets Formula = (=L3*(L3-1)*(L3-2)*(K3^3)+3*L3*(L3-1)*(K3^2)+L3*K3)48Google Sheets Function (All)
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49p(1-p)*(1-2p) (1-2p+2p^(2))⇒⇒0.065280.06534949HYPGEOMDIST(num_successes, num_draws, successes_in_pop, pop_size)
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50Google Sheets Formula = (=D3*(1-D3)*(1-2*D3)*(1-2*D3+2*D3^2))5050P{X=x} [=HYPGEOMDIST(x,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.000064499484
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51Moment Generating Function51Third Moment of X about the Mean = 51P{X≤x} [=HYPGEOMDIST(x,n,d,N)] Note: no cumulative in this function in Google Sheets.0.000064499484
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52M(t) = E[e^(tX)] = 1-p+pe^(t) 52E[(X-E[X])^(3)]= 52P{X<x} [=HYPGEOMDIST(x-1,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.0048374613
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531-p+pe^(t) 53np(1-p)*(1-2p) ⇒⇒⇒⇒-0.72-0.7253P{X>x} [=1-HYPGEOMDIST(x,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.000064499484
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5454Google Sheets Formula = (=(L3*K3)*(1-K3)*(1-2*K3))54P{X≥x} [=1-HYPGEOMDIST(x-1,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.9951625387
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55E[X] from MGF5555
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56E[X]= ∂{1-p+pe^(t)}/∂x |0 = p⇒⇒0.25656x<y
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5757Distribution Function57P{x<X<y} [=HYPGEOMDIST(y-1,n,d,N)] - [=HYPGEOMDIST(x,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.334190301
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58E[X^(2)] from MGF580............................................t<058P{x<X≤y} [=HYPGEOMDIST(y,n,d,N)] - [=HYPGEOMDIST(x,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.2172011208
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59E[X^(2)]= ∂^(2){1-p+pe^(t)}/∂x^(2) |0 = p⇒⇒0.259(nCx)*(p^(x))*(1-p)^(n-x) .....0 ≤t ≤ n59P{x≤X<y} [=HYPGEOMDIST(y-1,n,d,N)] - [=HYPGEOMDIST(x-1,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.3294173392
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60601..........................................n ≤ t 60P{x≤X≤y} [=HYPGEOMDIST(y,n,d,N)] - [=HYPGEOMDIST(x-1,n,d,N)]⇒⇒⇒⇒⇒⇒⇒0.212428159
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61E[X^(k)] from MGF61(t is integer portion of t) 61
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62E[X^(k)]= ∂^(k){1-p+pe^(t)}/∂x^(k) |0 = p⇒⇒0.26262
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6363Other Moments/Kurtosis/Etc: 63
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6464http://mathworld.wolfram.com/BinomialDistribution.html64
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6666To Plot Binomial Distribution: 66This distribution describes the experiment where elements are picked at random without
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6767To Plot Binomial Distrubtion use this link: 67replacement.
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6868http://www.wolframalpha.com/input/?i=binomial+distribution+%2810%2C+.50%2968
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69Definition6969
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70The Bernoulli distribution, named after the swiss mathematician Jacques Bernoulli (1654–1705),70Google Sheets Function (All) 70
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71 This distribution describes a probabilistic experiment where a trial has two possible outcomes, a success or a failure. 71BINOMDIST(num_successes, num_trials, prob_success, cumulative)71
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7272P{X=x} [=BINOMDIST(x,n,p,0)]⇒⇒⇒⇒0.18593784480.185972
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73Success = p73P{X≤x} [=BINOMDIST(x,n,p,1)]⇒⇒⇒⇒0.78272229430.782773Note:
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74Failure = 1-p or 'q'74P{X<x} [=BINOMDIST(x-1,n,p,1)]⇒⇒⇒⇒0.59678444960.596874
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75Limits =Both p and q are limited to range of 0 to 175P{X>x} [=1-BINOMDIST(x,n,p,1)]⇒⇒⇒⇒0.21727770570.217375
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76Other =independence76P{X≥x} [=1-BINOMDIST(x-1,n,p,1)]⇒⇒⇒⇒0.40321555040.403276
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79Examples:79P{x<X<y} [=BINOMDIST(y-1,n,p,1)-[=BINOMDIST(x,n,p,1)]79
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80–Probability of a head in a single coin flip80P{x<X≤y} [=BINOMDIST(y,n,p,1)-[=BINOMDIST(x,n,p,1)]80
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81–Probability of having a boy81P{x≤X<y} [=BINOMDIST(y-1,n,p,1)-[=BINOMDIST(x-1,n,p,1)]81
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–Probability of getting a raise82P{x≤X≤y} [=BINOMDIST(y,n,p,1)-[=BINOMDIST(x-1,n,p,1)]
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86Definition
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87A Binomial Distribution is a Bernoulli event that happens many times.
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89A Binomial distribution also only has a “success/failure” outcome, but it is repeated many times.
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90•Examples:
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91–Probability of getting 7 heads in 40 coin tosses
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92–Probability of observing 14 boys of 20 babieshttp://en.wikipedia.org/wiki/Hypergeometric_distribution
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93–Probability of 8 people getting a raise of 30 employees
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