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data_analysis:probability_distributions [2023/02/25 16:41] prgram [Negative Binomial] |
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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| 실패 r번, 성공 k번 | 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 $r = k - 1$. | ||
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| === 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-x = 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 ===== | ||
