R 이항분포 (Binomial Distribution) : dbinom()

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	d	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. ^^