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--- b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/06 23:43] – [e.g.,] hkimscil
+++ b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/13 08:59] (current) – [Binomial Distributions] hkimscil
@@ Line 73: / Line 73: @@
 ## rather than p * q^(r-1)
 dgeom(x = 0:n, prob = p)
-hist(dgeom(x = 0:n, prob = p))
+# hist(dgeom(x = 0:n, prob = p))
+barplot(dgeom(x=0:n, p))
 </code>
@@ Line 87: / Line 88: @@
 [29] 0.0003868563 0.0003094850
 >
-> hist(dgeom(x = 0:n, prob = p))
+> # hist(dgeom(x = 0:n, prob = p))
+> barplot(dgeom(x=0:n, p))
 </code>
-{{:b:head_first_statistics:pasted:20191030-023820.png}}
+<code> {{:b:head_first_statistics:pasted:20191030-023820.png}} </code>
+{{:b:head_first_statistics:pasted:20251013-080224.png}}
 r번 시도한 이후, 그 이후 어디서든지 간에 성공을 얻을 확률
 $$ P(X > r) = q^{r} $$
@@ Line 737: / Line 739: @@
 $Var(X) = \displaystyle \frac{q}{p^{2}}$
+<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
+> 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
+>
+</code>
@@ Line 787: / 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