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--- b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/13 08:28] – [e.g.,] hkimscil
+++ b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/13 08:59] (current) – [Binomial Distributions] hkimscil
@@ Line 740: / Line 740: @@
 <code>
+> p <- .4
+> q <- 1-p
+>
+> p*q^(2-1)
+[1] 0.24
 > dgeom(1, p)
 [1] 0.24
+>
+> 1-q^4
+[1] 0.8704
 > dgeom(0:3, p)
 [1] 0.4000 0.2400 0.1440 0.0864
 > sum(dgeom(0:3, p))
 [1] 0.8704
-> 1-q^4
+> pgeom(3, p)
 [1] 0.8704
+>
 > q^4
 [1] 0.1296
 > 1-sum(dgeom(0:3, p))
 [1] 0.1296
+> 1-pgeom(3, p)
+[1] 0.1296
+> pgeom(3, p, lower.tail = F)
+[1] 0.1296
+>
 > 1/p
 [1] 2.5
+>
 > q/p^2
 [1] 3.75
@@ Line 807: / Line 822: @@
 \begin{eqnarray*}
 P(X = r) & = & _{n}C_{r} \cdot p^{r} \cdot q^{n-r} \;\;\; \text{Where,} \\
-_{n}C_{r} & = & \frac {n!}{r!(n-r)!}
+\displaystyle _{n}C_{r} & = & \displaystyle \dfrac {n!}{r!(n-r)!} \\
+\text{c.f.,  } \\
+\displaystyle _{n} P_{r} & = & \displaystyle \dfrac {n!} {(n-r)!} \\
 \end{eqnarray*}
+see [[:b:head_first_statistics:permutation_and_combination#what_if_horse_order_doesn_t_matter|Permutation chapter]]
 p = 각 시행에서 성공할 확률