Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
data_analysis:probability_distributions [2023/02/25 16:38] prgram [how?] |
data_analysis:probability_distributions [2023/02/25 18:23] (current) prgram [Probability Distributions] |
||
---|---|---|---|
Line 1: | Line 1: | ||
====== Probability Distributions ====== | ====== Probability Distributions ====== | ||
+ | {{INLINETOC}} | ||
===== Discrete ===== | ===== Discrete ===== | ||
Line 24: | Line 24: | ||
==== 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-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 === | ||
Line 44: | Line 48: | ||
$$P(X = x) = p(1-p)^{x-1}$$ | $$P(X = x) = p(1-p)^{x-1}$$ | ||
+ | |||
+ | ==== Poisson ==== | ||
===== Continuous ===== | ===== Continuous ===== |