'베르누이 시행'에 해당되는 글 1건

  1. 2015.09.13 R 이항분포 (Binomial Distribution) : dbinom() (4)

통계에서 다양한 확률분포를 배웠을 텐데요, 왜 확률분포를 어디에 써먹는 것인지 잘 모르고 '어려운 개념'에 통계를 멀리하게 되는 시발점이 되는 경우가 많지 않았을까 싶습니다.  

 

만약 우리가 모집단이 어떤 확률분포를 띠고 있는지를 안다면 주어진 분위수에 대한 확률을 계산한다거나, 아니면 특정 확률에 해당하는 분위수를 계산할 수 있습니다.  그리고 시뮬레이션을 한다고 했을 때 확률분포를 이용해서 난수를 발생시켜서 사용할 수도 있습니다.

 

확률분포는 크게 이산형 확률 분포(Discrete probability distributon)과 연속형 확률 분포(Continuous probability distribution)으로 나눌 수 있습니다. 

 

이산형 확률 분포(Discrete Probability Distribution)에는 이항분포(Binomial distribution), 초기하분포(Hypergeometric distribution), 포아송 분포(Poisson distribution) 등이 있습니다. 

 

연속형 확률 분포 (Continuous probability distribution)에는 정규분포(Normal distributio), t-분포(t-distribution), F분포(F-distributio), 균등분포(Uniform distribution), 카이제곱분포(Chisq-distribution), 감마분포(Gamma distribution) 등이 있습니다.

 

이번 포스팅에서는 이산형 확률 분포 중 첫번째로 이항분포(Binomial distiribution)의 기본 개념에 대해 알아보고 R로 이항분포 그래프, 주어진 분위수에 대한 확률 계산, 그리고 특정 확률에 해당하는 분위수 계산, 난수 발생을 하는 방법에 대해 소개해 보겠습니다.

 

 

어떤 실험을 반복해서 시행한다고 했을 때 각 시행마다 "성공(success, 1)" 또는 "실패(failure, 0)"의 두 가지 경우의 수만 나온다고 할 때, 우리는 이런 시행을 "베르누이 시행(Bernoulli trial)"이라고 합니다. 

 

그리고 성공확률이 p인 베르누이 시행을 n번 반복했을 때 성공하는 횟수를 X라 하면, 확률변수 X는 모수 n과 p인 이항분포(Binomial distributio)을 따른다고 합니다. 

 

 

 

* 참고: f(x)의 첫번째 줄에 있는 n개 중에서 x개를 복원추출로 뽑는(다른 말로, n번 시행할 때 x번 성공하는) nCx = n! / x!(n-x)!

 

 

예를 들어서, 시행 회수 20회, 복원추출, 성공/실패 확률 50%인 베르누이 시행 (가령, 동전 던지기 앞(Head), 뒤(Tail) 을 R의 sample() 함수를 사용해서 시뮬레이션을 해보면 아래와 같습니다.  시행을 할 때마다 조금씩 달라지지만 성공/실패 회수는 거의 반, 반으로 비슷함을 알 수 있습니다.

 

 
> sample(c("H", "T"), size=20, replace=TRUE, prob=c(0.5, 0.5))
 [1] "H" "T" "T" "H" "H" "T" "H" "T" "H" "H" "H" "T" "H" "T" "H" "T" "T" "H" "H" "H"
> sample(c("H", "T"), size=20, replace=TRUE, prob=c(0.5, 0.5))
 [1] "T" "H" "T" "H" "H" "H" "H" "T" "H" "H" "H" "H" "T" "T" "H" "T" "T" "T" "H" "T"
> sample(c("H", "T"), size=20, replace=TRUE, prob=c(0.5, 0.5))
 [1] "H" "H" "T" "T" "H" "H" "H" "T" "T" "T" "T" "H" "H" "T" "T" "T" "T" "H" "T" "T"

 

 

 

이항분포의 밀도 함수, 누적 분포 함수, 분위수 함수, 난수 발생을 위한 R 함수 및 모수는 다음과 같습니다.

 

 구분

이항분포(binom) R 함수/모수 

  밀도 함수

  dbinom(x, size, prob)

  누적 분포 함수

 p

  pbinom(q, size, prob, lower.tail = TRUE / FALSE)

  분위수 함수

 q

  qbinom(p, size,  prob, lower.tail = TRUE / FALSE)

  난수 발생

 r

  rbinom(n, size, prob)

 

 

 

(1) 성공확률 0.5인 베르누이 시행을 20회 했을 때의 이항분포 밀도함수 : plot()

 

 
> y <- dbinom(0:20, size=20, prob=0.5)
> plot(0:20, y, type='h', lwd=5, col="grey", ylab="Probability", xlab="확률변수 X", 
+      main = c("X ~ B(20, 0.5)"))

 

 

 

 

 

 

(2) P(X = 12) 확률 계산 : dbinom(x, size, prob)

 

 

> # P(X = 12) 확률 계산
> dbinom(12, size=20, prob=0.5)
[1] 0.1201344
 

 

 

 

(3) P(X <= 12) 확률 계산 : pbinom(x, size, prob, lower.tail=TRUE)

 

 

> # (3) P(X <= 12) 확률 계산 > pbinom(12, size=20, prob=0.5, lower.tail = TRUE) [1] 0.868412 > > sum(dbinom(0:12, size=20, prob=0.5)) # dbinom()을 sum()해도 동일한 값 [1] 0.868412

 

 

 

(4) P(X > 12) 확률 계산 : pbinom(x, size, prob, lower.tail = FALSE)

 

 

> pbinom(12, size=20, prob=0.5, lower.tail = FALSE)
[1] 0.131588

> 1 - pbinom(12, size=20, prob=0.5, lower.tail = TRUE) # 동일한 값 [1] 0.131588

 

 

 

 

(5) 이항분포  난수 발생 : rbinom(n, size, prob)

 

> rbinom(12, size=20, prob=0.5)
 [1] 13 10  9 14 10  9  9  7 17  9  6 10
> 
> rbinom(12, size=20, prob=0.5)
 [1] 11 11  9 13  4 12  7 10 10 14 10 13
> 
> rbinom(12, size=20, prob=0.5)
 [1]  7  5 11  9 13 10 11 14 10 14  4 10

 

 

 

 

(6) 이항분포 누적분포함수 그래프

 

 

> # (6) 누적분포함수 그래프
> plot(pbinom(0:20, size=20, prob=0.5), type='h')
 

 

 

 

 

 

 

이메일로 아래와 같은 질문을 보내주신 분이 계셔서 질문이랑 답변 공유합니다.  repeat{} 로 프로그램 짜서 루프를 돌려서 근사값을 구했습니다.

 

Question) 

이항분포에서 prob의 값이 주어지지 않았을때 값을 구할 수 있을까요?

 

예를들어 200번 실험 중 50번 이상 실험이 설공할 확률이 0.2보다 클 경우를 구하려고 합니다.

1-pbinom(49,200,prob)>0.2

인데, 이와 관련해 prob를 구할 수 있을까요?

 

 

> i <- 0.01
> repeat {
+   pbinom_x_50_upper_size_200 <- pbinom(50, size=200, prob=i, lower.tail = FALSE) 
+   cat("pbinom(x=50, size=200, prob=", i, ", lower.tail=FALSE) = ", pbinom_x_50_upper_size_200, "\n", sep="")
+   if (pbinom_x_50_upper_size_200 > 0.2) break
+   i <- i+0.001
+ }
pbinom(x=50, size=200, prob=0.01, lower.tail=FALSE) = 3.0749e-55
pbinom(x=50, size=200, prob=0.011, lower.tail=FALSE) = 3.425864e-53
pbinom(x=50, size=200, prob=0.012, lower.tail=FALSE) = 2.499419e-51
pbinom(x=50, size=200, prob=0.013, lower.tail=FALSE) = 1.277906e-49
pbinom(x=50, size=200, prob=0.014, lower.tail=FALSE) = 4.826703e-48
pbinom(x=50, size=200, prob=0.015, lower.tail=FALSE) = 1.40426e-46
pbinom(x=50, size=200, prob=0.016, lower.tail=FALSE) = 3.254805e-45
pbinom(x=50, size=200, prob=0.017, lower.tail=FALSE) = 6.178075e-44
pbinom(x=50, size=200, prob=0.018, lower.tail=FALSE) = 9.825351e-43
pbinom(x=50, size=200, prob=0.019, lower.tail=FALSE) = 1.334491e-41
pbinom(x=50, size=200, prob=0.02, lower.tail=FALSE) = 1.573198e-40
pbinom(x=50, size=200, prob=0.021, lower.tail=FALSE) = 1.632106e-39
pbinom(x=50, size=200, prob=0.022, lower.tail=FALSE) = 1.507896e-38
pbinom(x=50, size=200, prob=0.023, lower.tail=FALSE) = 1.253499e-37
pbinom(x=50, size=200, prob=0.024, lower.tail=FALSE) = 9.460336e-37
pbinom(x=50, size=200, prob=0.025, lower.tail=FALSE) = 6.533438e-36
pbinom(x=50, size=200, prob=0.026, lower.tail=FALSE) = 4.157663e-35
pbinom(x=50, size=200, prob=0.027, lower.tail=FALSE) = 2.453052e-34
pbinom(x=50, size=200, prob=0.028, lower.tail=FALSE) = 1.349272e-33
pbinom(x=50, size=200, prob=0.029, lower.tail=FALSE) = 6.952835e-33
pbinom(x=50, size=200, prob=0.03, lower.tail=FALSE) = 3.371404e-32
pbinom(x=50, size=200, prob=0.031, lower.tail=FALSE) = 1.544453e-31
pbinom(x=50, size=200, prob=0.032, lower.tail=FALSE) = 6.708377e-31
pbinom(x=50, size=200, prob=0.033, lower.tail=FALSE) = 2.771777e-30
pbinom(x=50, size=200, prob=0.034, lower.tail=FALSE) = 1.092671e-29
pbinom(x=50, size=200, prob=0.035, lower.tail=FALSE) = 4.120884e-29
pbinom(x=50, size=200, prob=0.036, lower.tail=FALSE) = 1.490533e-28
pbinom(x=50, size=200, prob=0.037, lower.tail=FALSE) = 5.182438e-28
pbinom(x=50, size=200, prob=0.038, lower.tail=FALSE) = 1.735721e-27
pbinom(x=50, size=200, prob=0.039, lower.tail=FALSE) = 5.610737e-27
pbinom(x=50, size=200, prob=0.04, lower.tail=FALSE) = 1.753602e-26
pbinom(x=50, size=200, prob=0.041, lower.tail=FALSE) = 5.30802e-26
pbinom(x=50, size=200, prob=0.042, lower.tail=FALSE) = 1.558447e-25
pbinom(x=50, size=200, prob=0.043, lower.tail=FALSE) = 4.444567e-25
pbinom(x=50, size=200, prob=0.044, lower.tail=FALSE) = 1.232885e-24
pbinom(x=50, size=200, prob=0.045, lower.tail=FALSE) = 3.330502e-24
pbinom(x=50, size=200, prob=0.046, lower.tail=FALSE) = 8.771904e-24
pbinom(x=50, size=200, prob=0.047, lower.tail=FALSE) = 2.254996e-23
pbinom(x=50, size=200, prob=0.048, lower.tail=FALSE) = 5.663776e-23
pbinom(x=50, size=200, prob=0.049, lower.tail=FALSE) = 1.391196e-22
pbinom(x=50, size=200, prob=0.05, lower.tail=FALSE) = 3.344884e-22
pbinom(x=50, size=200, prob=0.051, lower.tail=FALSE) = 7.878612e-22
pbinom(x=50, size=200, prob=0.052, lower.tail=FALSE) = 1.819442e-21
pbinom(x=50, size=200, prob=0.053, lower.tail=FALSE) = 4.122599e-21
pbinom(x=50, size=200, prob=0.054, lower.tail=FALSE) = 9.171801e-21
pbinom(x=50, size=200, prob=0.055, lower.tail=FALSE) = 2.004832e-20
pbinom(x=50, size=200, prob=0.056, lower.tail=FALSE) = 4.308392e-20
pbinom(x=50, size=200, prob=0.057, lower.tail=FALSE) = 9.108046e-20
pbinom(x=50, size=200, prob=0.058, lower.tail=FALSE) = 1.895194e-19
pbinom(x=50, size=200, prob=0.059, lower.tail=FALSE) = 3.883594e-19
pbinom(x=50, size=200, prob=0.06, lower.tail=FALSE) = 7.841276e-19
pbinom(x=50, size=200, prob=0.061, lower.tail=FALSE) = 1.560713e-18
pbinom(x=50, size=200, prob=0.062, lower.tail=FALSE) = 3.063672e-18
pbinom(x=50, size=200, prob=0.063, lower.tail=FALSE) = 5.933835e-18
pbinom(x=50, size=200, prob=0.064, lower.tail=FALSE) = 1.134446e-17
pbinom(x=50, size=200, prob=0.065, lower.tail=FALSE) = 2.141708e-17
pbinom(x=50, size=200, prob=0.066, lower.tail=FALSE) = 3.994201e-17
pbinom(x=50, size=200, prob=0.067, lower.tail=FALSE) = 7.361232e-17
pbinom(x=50, size=200, prob=0.068, lower.tail=FALSE) = 1.341135e-16
pbinom(x=50, size=200, prob=0.069, lower.tail=FALSE) = 2.416241e-16
pbinom(x=50, size=200, prob=0.07, lower.tail=FALSE) = 4.306171e-16
pbinom(x=50, size=200, prob=0.071, lower.tail=FALSE) = 7.593772e-16
pbinom(x=50, size=200, prob=0.072, lower.tail=FALSE) = 1.325457e-15
pbinom(x=50, size=200, prob=0.073, lower.tail=FALSE) = 2.290532e-15
pbinom(x=50, size=200, prob=0.074, lower.tail=FALSE) = 3.920003e-15
pbinom(x=50, size=200, prob=0.075, lower.tail=FALSE) = 6.645489e-15
pbinom(x=50, size=200, prob=0.076, lower.tail=FALSE) = 1.11626e-14
pbinom(x=50, size=200, prob=0.077, lower.tail=FALSE) = 1.858251e-14
pbinom(x=50, size=200, prob=0.078, lower.tail=FALSE) = 3.066499e-14
pbinom(x=50, size=200, prob=0.079, lower.tail=FALSE) = 5.01737e-14
pbinom(x=50, size=200, prob=0.08, lower.tail=FALSE) = 8.141335e-14
pbinom(x=50, size=200, prob=0.081, lower.tail=FALSE) = 1.310357e-13
pbinom(x=50, size=200, prob=0.082, lower.tail=FALSE) = 2.092391e-13
pbinom(x=50, size=200, prob=0.083, lower.tail=FALSE) = 3.315409e-13
pbinom(x=50, size=200, prob=0.084, lower.tail=FALSE) = 5.213765e-13
pbinom(x=50, size=200, prob=0.085, lower.tail=FALSE) = 8.138827e-13
pbinom(x=50, size=200, prob=0.086, lower.tail=FALSE) = 1.261367e-12
pbinom(x=50, size=200, prob=0.087, lower.tail=FALSE) = 1.941161e-12
pbinom(x=50, size=200, prob=0.088, lower.tail=FALSE) = 2.966815e-12
pbinom(x=50, size=200, prob=0.089, lower.tail=FALSE) = 4.503954e-12
pbinom(x=50, size=200, prob=0.09, lower.tail=FALSE) = 6.792603e-12
pbinom(x=50, size=200, prob=0.091, lower.tail=FALSE) = 1.017839e-11
pbinom(x=50, size=200, prob=0.092, lower.tail=FALSE) = 1.51559e-11
pbinom(x=50, size=200, prob=0.093, lower.tail=FALSE) = 2.242863e-11
pbinom(x=50, size=200, prob=0.094, lower.tail=FALSE) = 3.299116e-11
pbinom(x=50, size=200, prob=0.095, lower.tail=FALSE) = 4.824148e-11
pbinom(x=50, size=200, prob=0.096, lower.tail=FALSE) = 7.01333e-11
pbinom(x=50, size=200, prob=0.097, lower.tail=FALSE) = 1.013817e-10
pbinom(x=50, size=200, prob=0.098, lower.tail=FALSE) = 1.457388e-10
pbinom(x=50, size=200, prob=0.099, lower.tail=FALSE) = 2.083625e-10
pbinom(x=50, size=200, prob=0.1, lower.tail=FALSE) = 2.963049e-10
pbinom(x=50, size=200, prob=0.101, lower.tail=FALSE) = 4.191581e-10
pbinom(x=50, size=200, prob=0.102, lower.tail=FALSE) = 5.899028e-10
pbinom(x=50, size=200, prob=0.103, lower.tail=FALSE) = 8.260166e-10
pbinom(x=50, size=200, prob=0.104, lower.tail=FALSE) = 1.150917e-09
pbinom(x=50, size=200, prob=0.105, lower.tail=FALSE) = 1.595829e-09
pbinom(x=50, size=200, prob=0.106, lower.tail=FALSE) = 2.202187e-09
pbinom(x=50, size=200, prob=0.107, lower.tail=FALSE) = 3.024722e-09
pbinom(x=50, size=200, prob=0.108, lower.tail=FALSE) = 4.135395e-09
pbinom(x=50, size=200, prob=0.109, lower.tail=FALSE) = 5.628391e-09
pbinom(x=50, size=200, prob=0.11, lower.tail=FALSE) = 7.626442e-09
pbinom(x=50, size=200, prob=0.111, lower.tail=FALSE) = 1.028878e-08
pbinom(x=50, size=200, prob=0.112, lower.tail=FALSE) = 1.382111e-08
pbinom(x=50, size=200, prob=0.113, lower.tail=FALSE) = 1.848804e-08
pbinom(x=50, size=200, prob=0.114, lower.tail=FALSE) = 2.462857e-08
pbinom(x=50, size=200, prob=0.115, lower.tail=FALSE) = 3.267517e-08
pbinom(x=50, size=200, prob=0.116, lower.tail=FALSE) = 4.317741e-08
pbinom(x=50, size=200, prob=0.117, lower.tail=FALSE) = 5.683084e-08
pbinom(x=50, size=200, prob=0.118, lower.tail=FALSE) = 7.451237e-08
pbinom(x=50, size=200, prob=0.119, lower.tail=FALSE) = 9.732331e-08
pbinom(x=50, size=200, prob=0.12, lower.tail=FALSE) = 1.266415e-07
pbinom(x=50, size=200, prob=0.121, lower.tail=FALSE) = 1.641845e-07
pbinom(x=50, size=200, prob=0.122, lower.tail=FALSE) = 2.120853e-07
pbinom(x=50, size=200, prob=0.123, lower.tail=FALSE) = 2.729831e-07
pbinom(x=50, size=200, prob=0.124, lower.tail=FALSE) = 3.501321e-07
pbinom(x=50, size=200, prob=0.125, lower.tail=FALSE) = 4.475306e-07
pbinom(x=50, size=200, prob=0.126, lower.tail=FALSE) = 5.700739e-07
pbinom(x=50, size=200, prob=0.127, lower.tail=FALSE) = 7.237351e-07
pbinom(x=50, size=200, prob=0.128, lower.tail=FALSE) = 9.157779e-07
pbinom(x=50, size=200, prob=0.129, lower.tail=FALSE) = 1.155006e-06
pbinom(x=50, size=200, prob=0.13, lower.tail=FALSE) = 1.452053e-06
pbinom(x=50, size=200, prob=0.131, lower.tail=FALSE) = 1.819725e-06
pbinom(x=50, size=200, prob=0.132, lower.tail=FALSE) = 2.273389e-06
pbinom(x=50, size=200, prob=0.133, lower.tail=FALSE) = 2.831432e-06
pbinom(x=50, size=200, prob=0.134, lower.tail=FALSE) = 3.515784e-06
pbinom(x=50, size=200, prob=0.135, lower.tail=FALSE) = 4.352516e-06
pbinom(x=50, size=200, prob=0.136, lower.tail=FALSE) = 5.372534e-06
pbinom(x=50, size=200, prob=0.137, lower.tail=FALSE) = 6.612358e-06
pbinom(x=50, size=200, prob=0.138, lower.tail=FALSE) = 8.115017e-06
pbinom(x=50, size=200, prob=0.139, lower.tail=FALSE) = 9.93106e-06
pbinom(x=50, size=200, prob=0.14, lower.tail=FALSE) = 1.211969e-05
pbinom(x=50, size=200, prob=0.141, lower.tail=FALSE) = 1.475007e-05
pbinom(x=50, size=200, prob=0.142, lower.tail=FALSE) = 1.79027e-05
pbinom(x=50, size=200, prob=0.143, lower.tail=FALSE) = 2.167112e-05
pbinom(x=50, size=200, prob=0.144, lower.tail=FALSE) = 2.616362e-05
pbinom(x=50, size=200, prob=0.145, lower.tail=FALSE) = 3.150524e-05
pbinom(x=50, size=200, prob=0.146, lower.tail=FALSE) = 3.784002e-05
pbinom(x=50, size=200, prob=0.147, lower.tail=FALSE) = 4.533335e-05
pbinom(x=50, size=200, prob=0.148, lower.tail=FALSE) = 5.417471e-05
pbinom(x=50, size=200, prob=0.149, lower.tail=FALSE) = 6.458051e-05
pbinom(x=50, size=200, prob=0.15, lower.tail=FALSE) = 7.679736e-05
pbinom(x=50, size=200, prob=0.151, lower.tail=FALSE) = 9.110546e-05
pbinom(x=50, size=200, prob=0.152, lower.tail=FALSE) = 0.0001078224
pbinom(x=50, size=200, prob=0.153, lower.tail=FALSE) = 0.0001273072
pbinom(x=50, size=200, prob=0.154, lower.tail=FALSE) = 0.0001499647
pbinom(x=50, size=200, prob=0.155, lower.tail=FALSE) = 0.0001762501
pbinom(x=50, size=200, prob=0.156, lower.tail=FALSE) = 0.0002066743
pbinom(x=50, size=200, prob=0.157, lower.tail=FALSE) = 0.0002418089
pbinom(x=50, size=200, prob=0.158, lower.tail=FALSE) = 0.000282292
pbinom(x=50, size=200, prob=0.159, lower.tail=FALSE) = 0.0003288341
pbinom(x=50, size=200, prob=0.16, lower.tail=FALSE) = 0.0003822243
pbinom(x=50, size=200, prob=0.161, lower.tail=FALSE) = 0.0004433374
pbinom(x=50, size=200, prob=0.162, lower.tail=FALSE) = 0.0005131399
pbinom(x=50, size=200, prob=0.163, lower.tail=FALSE) = 0.0005926979
pbinom(x=50, size=200, prob=0.164, lower.tail=FALSE) = 0.0006831842
pbinom(x=50, size=200, prob=0.165, lower.tail=FALSE) = 0.0007858856
pbinom(x=50, size=200, prob=0.166, lower.tail=FALSE) = 0.0009022113
pbinom(x=50, size=200, prob=0.167, lower.tail=FALSE) = 0.0010337
pbinom(x=50, size=200, prob=0.168, lower.tail=FALSE) = 0.001182029
pbinom(x=50, size=200, prob=0.169, lower.tail=FALSE) = 0.001349022
pbinom(x=50, size=200, prob=0.17, lower.tail=FALSE) = 0.001536655
pbinom(x=50, size=200, prob=0.171, lower.tail=FALSE) = 0.001747067
pbinom(x=50, size=200, prob=0.172, lower.tail=FALSE) = 0.001982569
pbinom(x=50, size=200, prob=0.173, lower.tail=FALSE) = 0.002245645
pbinom(x=50, size=200, prob=0.174, lower.tail=FALSE) = 0.00253897
pbinom(x=50, size=200, prob=0.175, lower.tail=FALSE) = 0.002865405
pbinom(x=50, size=200, prob=0.176, lower.tail=FALSE) = 0.003228015
pbinom(x=50, size=200, prob=0.177, lower.tail=FALSE) = 0.003630065
pbinom(x=50, size=200, prob=0.178, lower.tail=FALSE) = 0.004075034
pbinom(x=50, size=200, prob=0.179, lower.tail=FALSE) = 0.004566614
pbinom(x=50, size=200, prob=0.18, lower.tail=FALSE) = 0.005108717
pbinom(x=50, size=200, prob=0.181, lower.tail=FALSE) = 0.005705476
pbinom(x=50, size=200, prob=0.182, lower.tail=FALSE) = 0.006361248
pbinom(x=50, size=200, prob=0.183, lower.tail=FALSE) = 0.007080619
pbinom(x=50, size=200, prob=0.184, lower.tail=FALSE) = 0.007868395
pbinom(x=50, size=200, prob=0.185, lower.tail=FALSE) = 0.00872961
pbinom(x=50, size=200, prob=0.186, lower.tail=FALSE) = 0.009669517
pbinom(x=50, size=200, prob=0.187, lower.tail=FALSE) = 0.01069359
pbinom(x=50, size=200, prob=0.188, lower.tail=FALSE) = 0.0118075
pbinom(x=50, size=200, prob=0.189, lower.tail=FALSE) = 0.01301716
pbinom(x=50, size=200, prob=0.19, lower.tail=FALSE) = 0.01432863
pbinom(x=50, size=200, prob=0.191, lower.tail=FALSE) = 0.01574819
pbinom(x=50, size=200, prob=0.192, lower.tail=FALSE) = 0.01728228
pbinom(x=50, size=200, prob=0.193, lower.tail=FALSE) = 0.01893752
pbinom(x=50, size=200, prob=0.194, lower.tail=FALSE) = 0.02072064
pbinom(x=50, size=200, prob=0.195, lower.tail=FALSE) = 0.02263853
pbinom(x=50, size=200, prob=0.196, lower.tail=FALSE) = 0.02469818
pbinom(x=50, size=200, prob=0.197, lower.tail=FALSE) = 0.02690666
pbinom(x=50, size=200, prob=0.198, lower.tail=FALSE) = 0.02927113
pbinom(x=50, size=200, prob=0.199, lower.tail=FALSE) = 0.03179876
pbinom(x=50, size=200, prob=0.2, lower.tail=FALSE) = 0.03449677
pbinom(x=50, size=200, prob=0.201, lower.tail=FALSE) = 0.03737237
pbinom(x=50, size=200, prob=0.202, lower.tail=FALSE) = 0.0404327
pbinom(x=50, size=200, prob=0.203, lower.tail=FALSE) = 0.04368489
pbinom(x=50, size=200, prob=0.204, lower.tail=FALSE) = 0.04713593
pbinom(x=50, size=200, prob=0.205, lower.tail=FALSE) = 0.0507927
pbinom(x=50, size=200, prob=0.206, lower.tail=FALSE) = 0.05466193
pbinom(x=50, size=200, prob=0.207, lower.tail=FALSE) = 0.05875016
pbinom(x=50, size=200, prob=0.208, lower.tail=FALSE) = 0.06306369
pbinom(x=50, size=200, prob=0.209, lower.tail=FALSE) = 0.0676086
pbinom(x=50, size=200, prob=0.21, lower.tail=FALSE) = 0.07239063
pbinom(x=50, size=200, prob=0.211, lower.tail=FALSE) = 0.07741525
pbinom(x=50, size=200, prob=0.212, lower.tail=FALSE) = 0.08268754
pbinom(x=50, size=200, prob=0.213, lower.tail=FALSE) = 0.08821221
pbinom(x=50, size=200, prob=0.214, lower.tail=FALSE) = 0.09399354
pbinom(x=50, size=200, prob=0.215, lower.tail=FALSE) = 0.1000354
pbinom(x=50, size=200, prob=0.216, lower.tail=FALSE) = 0.1063411
pbinom(x=50, size=200, prob=0.217, lower.tail=FALSE) = 0.1129135
pbinom(x=50, size=200, prob=0.218, lower.tail=FALSE) = 0.1197549
pbinom(x=50, size=200, prob=0.219, lower.tail=FALSE) = 0.1268671
pbinom(x=50, size=200, prob=0.22, lower.tail=FALSE) = 0.1342513
pbinom(x=50, size=200, prob=0.221, lower.tail=FALSE) = 0.141908
pbinom(x=50, size=200, prob=0.222, lower.tail=FALSE) = 0.1498372
pbinom(x=50, size=200, prob=0.223, lower.tail=FALSE) = 0.1580383
pbinom(x=50, size=200, prob=0.224, lower.tail=FALSE) = 0.1665098
pbinom(x=50, size=200, prob=0.225, lower.tail=FALSE) = 0.1752498
pbinom(x=50, size=200, prob=0.226, lower.tail=FALSE) = 0.1842556
pbinom(x=50, size=200, prob=0.227, lower.tail=FALSE) = 0.193524
pbinom(x=50, size=200, prob=0.228, lower.tail=FALSE) = 0.2030508

 

많은 도움 되었기를 바랍니다.

 

이번 포스팅이 도움이 되었다면 아래의 '공감 ~♡'를 꾸욱 눌러주세요. ^^

 

 

 

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