data_analysis:probability_distributions

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data_analysis:probability_distributions [2023/02/25 07:32] – prgramdata_analysis:probability_distributions [2025/07/07 14:12] (current) – external edit 127.0.0.1
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 ====== Probability Distributions ====== ====== Probability Distributions ======
 +{{INLINETOC}}
 ===== Discrete ===== ===== Discrete =====
  
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 ==== Negative Binomial ==== ==== Negative Binomial ====
 probability of the number of failures $r$ before observing $k$ successes in a sequence of independent and identically distributed Bernoulli trials.  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 $r = k - 1$. $X \sim \text{NegBin}(k, p)$ or $X \sim \text{NegBin}(r, p)$, depending on the parameterization used. The two parameterizations are related by the identity $r = k - 1$.
  
-$$P(X = k) = {k+r-1 \choose k} p^k (1-p)^r$$+$$P(X = k) = {k+r-1 \choose r} p^k (1-p)^r$$
  
 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. 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? === === how? ===
-y번실험 시도에 x번 성공 -> y-1번째까지 x-1번 성공 & y번째 성공 +** 성공횟수에 초점** 
-$$P(Y=y) = {y-1 \choose x-1} p^{x-1} (1-p)^{y-x} p $$ +m번실험 시도에 n번 성공 -> m-1번째까지 n-1번 성공 & m번째 성공 
-$x=r$ (실패횟수) -> $y=r+k$ -> $ y-k = k $+$$P(Y=n) = {m-1 \choose n-1} p^{n-1} (1-p)^{m-n} p $$ 
 +$n=k$ (성공횟수) -> $m=r+k$ -> $ m-n = r $ 
 +$ {k+r-1 \choose r} = {k+r-1 \choose k-1} $
  
 === example === === example ===
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 $$P(X = x) = p(1-p)^{x-1}$$ $$P(X = x) = p(1-p)^{x-1}$$
  
 +
 +==== Poisson ====
  
 ===== Continuous ===== ===== Continuous =====
  • Last modified: 2025/07/07 14:12