常见概率分布总结

最新推荐文章于 2025-04-28 07:51:30 发布

原创

最新推荐文章于 2025-04-28 07:51:30 发布 · 1k 阅读

CC 4.0 BY-SA版权

文章标签：

本文总结了常见的概率分布，包括离散分布如伯努利、二项、几何、负二项、超几何和泊松分布，以及连续分布如均匀、指数、伽马和正态分布。详细介绍了每个分布的概率质量函数、累积分布函数及其性质，如期望和方差，并提到了指数族分布的重要角色。

Discrete

pmf
- $f_X(x) = P(X= x) =\left\{\begin{aligned}(1-p)^{1-x}p^x & \quad \text{for x = 0 or 1}\\ 0 & \quad\text{otherwise}\end{aligned}\right.$
expectation
- $E (X) = p$

pmf
- $f_X(k) = P(X= k) =\left\{\begin{aligned}C_n^kp^k(1-p)^{n-k} & \quad \text{for k=0,1,....,n}\\ 0 & \quad\text{otherwise}\end{aligned}\right.$
expectation
- $E (X) = n p$
variance
- $v a r (X) = n p (1 - p)$

pmf
- $f_X(k) = P(X= k) =\left\{\begin{aligned}p(1-p)^{k-1} & \quad \text{for k=1,2,3...}\\ 0 & \quad\text{otherwise}\end{aligned}\right.$
expectation
- $\frac{1}{P}$

The negative binomial distribution arises as a generalization of the geometric distribution.
Suppose that a sequence of independent trials each with probability of success $p$ is performed until there are $r$ successes in all.
- so can be denote as $\cdot C_{k-1}^{r-1} p^{r-1}(1-p)^{(k-1)-(r-1)}$
pmf
- $f_X(k) = P(X= k) =\left\{\begin{aligned}C_{k-1}^{r-1}p^r(1-p)^{k-r} & \quad \text{for k=1,2,3...}\\ 0 & \quad\text{otherwise}\end{aligned}\right.$