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--- b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/06 23:40] – [e.g.,] hkimscil
+++ b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/06 23:43] (current) – [e.g.,] hkimscil
@@ Line 1022: / Line 1022: @@
 > </code>
-Q. 한 문제를 맞힐 확률은 1/4 이다. 총 여섯 문제가 있다고 할 때, 0에서 5 문제를 맞힐 확률은?
+Q. 한 문제를 맞힐 확률은 1/4 이다. 총 여섯 문제가 있다고 할 때, 0에서 5 문제를 맞힐 확률은? dbinom을 이용해서 구하시오.
 <code>
 p <- 1/4
@@ Line 1028: / Line 1028: @@
 n <- 6
 pbinom(5, n, p)
+- dbinom(6, n, p)
 </code>
+<code>
+> p <- 1/4
+> q <- 1-p
+> n <- 6
+> pbinom(5, n, p)
+[1] 0.9997559
+> 1 - dbinom(6, n, p)
+[1] 0.9997559
+</code>
+중요 . . . .
+<code>
+# http://commres.net/wiki/mean_and_variance_of_binomial_distribution
+# ##################################################################
+#
+p <- 1/4
+q <- 1 - p
+n <- 5
+r <- 0
+all.dens <- dbinom(0:n, n, p)
+all.dens
+sum(all.dens)
+choose(5,0)*p^0*(q^(5-0))
+choose(5,1)*p^1*(q^(5-1))
+choose(5,2)*p^2*(q^(5-2))
+choose(5,3)*p^3*(q^(5-3))
+choose(5,4)*p^4*(q^(5-4))
+choose(5,5)*p^5*(q^(5-5))
+all.dens
+choose(5,0)*p^0*(q^(5-0)) +
+  choose(5,1)*p^1*(q^(5-1)) +
+  choose(5,2)*p^2*(q^(5-2)) +
+  choose(5,3)*p^3*(q^(5-3)) +
+  choose(5,4)*p^4*(q^(5-4)) +
+  choose(5,5)*p^5*(q^(5-5))
+sum(all.dens)
+#
+(p+q)^n
+# note that n = whatever, (p+q)^n = 1
+</code>
+<code>
+> # http://commres.net/wiki/mean_and_variance_of_binomial_distribution
+> # ##################################################################
+> #
+> p <- 1/4
+> q <- 1 - p
+> n <- 5
+> r <- 0
+> all.dens <- dbinom(0:n, n, p)
+> all.dens
+[1] 0.2373046875 0.3955078125 0.2636718750 0.0878906250
+[5] 0.0146484375 0.0009765625
+> sum(all.dens)
+[1] 1
+>
+> choose(5,0)*p^0*(q^(5-0))
+[1] 0.2373047
+> choose(5,1)*p^1*(q^(5-1))
+[1] 0.3955078
+> choose(5,2)*p^2*(q^(5-2))
+[1] 0.2636719
+> choose(5,3)*p^3*(q^(5-3))
+[1] 0.08789062
+> choose(5,4)*p^4*(q^(5-4))
+[1] 0.01464844
+> choose(5,5)*p^5*(q^(5-5))
+[1] 0.0009765625
+> all.dens
+[1] 0.2373046875 0.3955078125 0.2636718750 0.0878906250
+[5] 0.0146484375 0.0009765625
+>
+> choose(5,0)*p^0*(q^(5-0)) +
++   choose(5,1)*p^1*(q^(5-1)) +
++   choose(5,2)*p^2*(q^(5-2)) +
++   choose(5,3)*p^3*(q^(5-3)) +
++   choose(5,4)*p^4*(q^(5-4)) +
++   choose(5,5)*p^5*(q^(5-5))
+[1] 1
+> sum(all.dens)
+[1] 1
+> #
+> (p+q)^n
+[1] 1
+> # note that n = whatever, (p+q)^n = 1
+>
+</code>
 ===== Proof of Binomial Expected Value and Variance =====
 [[:Mean and Variance of Binomial Distribution|이항분포에서의 기댓값과 분산에 대한 수학적 증명]], Mathematical proof of Binomial Distribution Expected value and Variance