> # Binomial in R
> (factorial(10)/(factorial(2)*factorial(8)))*(1/6)^2*(5/6)^8
[1] 0.29071
> dbinom(2, size=10, prob=1/6)
[1] 0.29071
> n<-10
> p<-1/6
> a<-seq(0,n)
> cbind(a,dbinom(a,n,p))
       a             
 [1,]  0 1.615056e-01
 [2,]  1 3.230112e-01
 [3,]  2 2.907100e-01
 [4,]  3 1.550454e-01
 [5,]  4 5.426588e-02
 [6,]  5 1.302381e-02
 [7,]  6 2.170635e-03
 [8,]  7 2.480726e-04
 [9,]  8 1.860544e-05
[10,]  9 8.269086e-07
[11,] 10 1.653817e-08

> options(scipen=999)
> cbind(a,dbinom(a,n,p))
       a                 
 [1,]  0 0.16150558288985
 [2,]  1 0.32301116577969
 [3,]  2 0.29071004920172
 [4,]  3 0.15504535957425
 [5,]  4 0.05426587585099
 [6,]  5 0.01302381020424
 [7,]  6 0.00217063503404
 [8,]  7 0.00024807257532
 [9,]  8 0.00001860544315
[10,]  9 0.00000082690858
[11,] 10 0.00000001653817

> cbind(a,round(dbinom(a,n,p),digits=4))
       a       
 [1,]  0 0.1615
 [2,]  1 0.3230
 [3,]  2 0.2907
 [4,]  3 0.1550
 [5,]  4 0.0543
 [6,]  5 0.0130
 [7,]  6 0.0022
 [8,]  7 0.0002
 [9,]  8 0.0000
[10,]  9 0.0000
[11,] 10 0.0000

> # Calculate CDF
> pbinom(2,10,1/6)
[1] 0.77523
> cbind(a,round(pbinom(a,n,p),digits=4))
       a       
 [1,]  0 0.1615
 [2,]  1 0.4845
 [3,]  2 0.7752
 [4,]  3 0.9303
 [5,]  4 0.9845
 [6,]  5 0.9976
 [7,]  6 0.9997
 [8,]  7 1.0000
 [9,]  8 1.0000
[10,]  9 1.0000
[11,] 10 1.0000

> # Calculate percentile
> qbinom(.90,10,1/6)
[1] 3
> rbinom(5,10,1/6)
[1] 4 2 1 1 3
> rbinom(5,10,1/6)
[1] 1 2 3 1 0

> # Hypergeometric
> dhyper(2,13,39,5)
[1] 0.27428
> b<-seq(0,5)
> cbind(b,round(dhyper(b,13,39,5),digits=4))
     b       
[1,] 0 0.2215
[2,] 1 0.4114
[3,] 2 0.2743
[4,] 3 0.0815
[5,] 4 0.0107
[6,] 5 0.0005
> cbind(b,round(phyper(b,13,39,5),digits=4))
     b       
[1,] 0 0.2215
[2,] 1 0.6330
[3,] 2 0.9072
[4,] 3 0.9888
[5,] 4 0.9995
[6,] 5 1.0000

> # Binomial approx
> cbind(b,round(dhyper(b,13,39,5),digits=4),round(dbinom(b,size=5,prob=1/4),digits=4))
     b              
[1,] 0 0.2215 0.2373
[2,] 1 0.4114 0.3955
[3,] 2 0.2743 0.2637
[4,] 3 0.0815 0.0879
[5,] 4 0.0107 0.0146
[6,] 5 0.0005 0.0010

> # Poisson
> dpois(2,lambda=2.5)
[1] 0.25652
> cbind(a,round(dpois(a,2.5),digits=4))
       a       
 [1,]  0 0.0821
 [2,]  1 0.2052
 [3,]  2 0.2565
 [4,]  3 0.2138
 [5,]  4 0.1336
 [6,]  5 0.0668
 [7,]  6 0.0278
 [8,]  7 0.0099
 [9,]  8 0.0031
[10,]  9 0.0009
[11,] 10 0.0002

> cbind(a,round(dpois(a,2.5),digits=4),round(dbinom(a,1000,.0025),digits=4),round(dbinom(a,100,.025),digits=4))
       a                     
 [1,]  0 0.0821 0.0818 0.0795
 [2,]  1 0.2052 0.2051 0.2039
 [3,]  2 0.2565 0.2567 0.2588
 [4,]  3 0.2138 0.2141 0.2168
 [5,]  4 0.1336 0.1337 0.1348
 [6,]  5 0.0668 0.0668 0.0664
 [7,]  6 0.0278 0.0277 0.0269
 [8,]  7 0.0099 0.0099 0.0093
 [9,]  8 0.0031 0.0031 0.0028
[10,]  9 0.0009 0.0008 0.0007
[11,] 10 0.0002 0.0002 0.0002