b:head_first_statistics:geometric_binomial_and_poisson_distributions
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| b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/06 14:43] – [e.g.,] hkimscil | b:head_first_statistics:geometric_binomial_and_poisson_distributions [2025/10/14 23:31] (current) – [e.g.,] hkimscil | ||
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| Line 73: | Line 73: | ||
| ## rather than p * q^(r-1) | ## rather than p * q^(r-1) | ||
| dgeom(x = 0:n, prob = p) | dgeom(x = 0:n, prob = p) | ||
| - | hist(dgeom(x = 0:n, prob = p)) | + | # hist(dgeom(x = 0:n, prob = p)) |
| + | barplot(dgeom(x=0: | ||
| </ | </ | ||
| Line 87: | Line 88: | ||
| [29] 0.0003868563 0.0003094850 | [29] 0.0003868563 0.0003094850 | ||
| > | > | ||
| - | > hist(dgeom(x = 0:n, prob = p)) | + | > # hist(dgeom(x = 0:n, prob = p)) |
| + | > barplot(dgeom(x=0: | ||
| </ | </ | ||
| - | {{: | + | < |
| + | {{: | ||
| r번 시도한 이후, 그 이후 어디서든지 간에 성공을 얻을 확률 | r번 시도한 이후, 그 이후 어디서든지 간에 성공을 얻을 확률 | ||
| $$ P(X > r) = q^{r} $$ | $$ P(X > r) = q^{r} $$ | ||
| Line 737: | Line 739: | ||
| $Var(X) = \displaystyle \frac{q}{p^{2}}$ | $Var(X) = \displaystyle \frac{q}{p^{2}}$ | ||
| + | < | ||
| + | > 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: | ||
| + | [1] 0.8704 | ||
| + | > pgeom(3, p) | ||
| + | [1] 0.8704 | ||
| + | > | ||
| + | > q^4 | ||
| + | [1] 0.1296 | ||
| + | > 1-sum(dgeom(0: | ||
| + | [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 787: | Line 822: | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| P(X = r) & = & _{n}C_{r} \cdot p^{r} \cdot q^{n-r} \;\;\; \text{Where, | P(X = r) & = & _{n}C_{r} \cdot p^{r} \cdot q^{n-r} \;\;\; \text{Where, | ||
| - | _{n}C_{r} & = & \frac {n!}{r!(n-r)!} | + | \displaystyle |
| + | \text{c.f., | ||
| + | \displaystyle _{n} P_{r} & = & \displaystyle \dfrac {n!} {(n-r)!} \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | |||
| + | see [[: | ||
| + | |||
| p = 각 시행에서 성공할 확률 | p = 각 시행에서 성공할 확률 | ||
| Line 882: | Line 922: | ||
| dbinom(r, n, p) | dbinom(r, n, p) | ||
| + | # dbinom(2, 5, 1/4) | ||
| </ | </ | ||
| Line 904: | Line 945: | ||
| > | > | ||
| </ | </ | ||
| - | |||
| - | |||
| - | |||
| - | |||
| - | |||
| Ans 2. | Ans 2. | ||
| Line 1028: | Line 1064: | ||
| n <- 6 | n <- 6 | ||
| pbinom(5, n, p) | pbinom(5, n, p) | ||
| - | |||
| 1 - dbinom(6, n, p) | 1 - dbinom(6, n, p) | ||
| + | sum(dbinom(0: | ||
| </ | </ | ||
| < | < | ||
| Line 1039: | Line 1075: | ||
| > 1 - dbinom(6, n, p) | > 1 - dbinom(6, n, p) | ||
| [1] 0.9997559 | [1] 0.9997559 | ||
| + | > sum(dbinom(0: | ||
| + | [1] 0.9997559 | ||
| + | > | ||
| </ | </ | ||
b/head_first_statistics/geometric_binomial_and_poisson_distributions.1759761812.txt.gz · Last modified: by hkimscil