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data_analysis:probability_distributions [2023/02/25 15:58]
prgram created 		data_analysis:probability_distributions [2023/02/25 18:23] (current)
prgram [Probability Distributions]
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 ====== Probability Distributions ====== ====== Probability Distributions ======
 +{{INLINETOC}}
 ===== Discrete ===== ===== Discrete =====
  
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 === Binomial과 관계 === === Binomial과 관계 ===
-$$ N +https://​m.blog.naver.com/​mykepzzang/​220839789774 
 +$ N \rightarrow \infty ​$ 면 Binomial 
 +$ X, Y \sim^\text{indep} \text{Bin} $ 일 때 $ P( X | X+Y ) $ (조건부확률) 은 Hypergeometric 
 +$ ( X \sim Bin(n,k), Y \sim Bin(N-n, K-k) ) $ 
 + 
 + 
 + 
 +==== Negative Binomial ==== 
 +probability of the number of failures $r$ before observing $k$ successes in a sequence of independent and identically distributed Bernoulli trials.  
 +실패 r번, 성공 k번 
 + 
 +** 실패횟수에 초점 ** 
 +$X \sim \text{NegBin}(k,​ p)$ or $X \sim \text{NegBin}(r,​ p)$, depending on the parameterization used. The two parameterizations are related by the identity $r = k - 1$. 
 + 
 +$$P(X = k) = {k+r-1 \choose r} p^k (1-p)^r$$ 
 + 
 +where $X$ is the random variable representing the number of trials until the $k$th success is observed, $p$ is the probability of success in a single trial, and $r$ is the number of failures before observing the $k$th success. 
 + 
 +=== how? === 
 +** 성공횟수에 초점** 
 +m번실험 시도에 n번 성공 -> m-1번째까지 n-1번 성공 & m번째 성공 
 +$$P(Y=n) = {m-1 \choose n-1} p^{n-1} (1-p)^{m-n} p $$ 
 +$n=k$ (성공횟수) -> $m=r+k$ -> $ m-n = r $ 
 +$ {k+r-1 \choose r} = {k+r-1 \choose k-1} $ 
 + 
 +=== example === 
 +For example, the negative binomial distribution can be used to model the number of phone calls a telemarketer must make before getting a fixed number of sales, the number of defective products produced before a certain number of acceptable products are produced, or the number of attempts required to solve a particular puzzle or problem. The negative binomial distribution is also used in some areas of biology to model the distribution of offspring from a single parent in a population. 
 + 
 + 
 +==== Geometric ==== 
 +Negative Binomial에서 k=1 일때 -> 성공할 때까지의 실험회수 (r=x-1) 
 +$$P(X = x) = p(1-p)^{x-1}$$
  
  
 +==== Poisson ====
  
 ===== Continuous ===== ===== Continuous =====