Probability Distributions

probability of $k$ successes in a sample of size $n$ drawn without replacement from a finite population of size $N$ that contains exactly $K$ successes.

N개 중 K개의 event가 있는 모집단에서, n개의 샘플을 뽑았을 때 event 가 k번 일어나는 확률

$X \sim \text{Hypergeom}(N, K, n)$
$$P(X = k) = \frac{{K \choose k}{N-K \choose n-k}}{N \choose n}$$

https://m.blog.naver.com/mykepzzang/220839789774
$ N \rightarrow \infty $ 면 Binomial
$ X, Y \sim^\text{indep} \text{Bin} $ 일 때 $ P( X | X+Y ) $ (조건부확률) 은 Hypergeometric
$ ( X \sim Bin(n,k), Y \sim Bin(N-n, K-k) ) $

probability of the number of failures $r$ before observing $k$ successes in a sequence of independent and identically distributed Bernoulli trials.
실패 r번, 성공 k번

실패횟수에 초점
$X \sim \text{NegBin}(k, p)$ or $X \sim \text{NegBin}(r, p)$, depending on the parameterization used. The two parameterizations are related by the identity $r = k - 1$.

$$P(X = k) = {k+r-1 \choose r} p^k (1-p)^r$$

where $X$ is the random variable representing the number of trials until the $k$th success is observed, $p$ is the probability of success in a single trial, and $r$ is the number of failures before observing the $k$th success.

성공횟수에 초점
m번실험 시도에 n번 성공 → m-1번째까지 n-1번 성공 & m번째 성공
$$P(Y=n) = {m-1 \choose n-1} p^{n-1} (1-p)^{m-n} p $$
$n=k$ (성공횟수) → $m=r+k$ → $ m-n = r $
$ {k+r-1 \choose r} = {k+r-1 \choose k-1} $

For example, the negative binomial distribution can be used to model the number of phone calls a telemarketer must make before getting a fixed number of sales, the number of defective products produced before a certain number of acceptable products are produced, or the number of attempts required to solve a particular puzzle or problem. The negative binomial distribution is also used in some areas of biology to model the distribution of offspring from a single parent in a population.

Negative Binomial에서 k=1 일때 → 성공할 때까지의 실험회수 (r=x-1)
$$P(X = x) = p(1-p)^{x-1}$$