二项分布与正态分布

随机变量的概率分布在统计学中占有重要地位。在这些概率分布中，二项分布和正态分布是现实生活中最常见的两种。

什么是二项分布？

二项式分布是与随机变量X相对应的概率分布，它是一系列独立的是/否实验的成功次数，每个实验都有一个成功的概率p。从X的定义来看，它显然是一个离散的随机变量；因此，二项分布也是离散的。

The distribution is denoted as X ~B(n,p) where n is the number of experiments and p is the probability of success. According to probability theory, we can deduce that B(n,p) follows the probability mass function . From this equation, it can be further deduced that the expected value of X, E(X) = np and the variance of X, V(X) = np(1-p).

For example, c***ider a random experiment of tossing a coin 3 times. Define success as obtaining H, failure as obtaining T and the random variable X as the number of successes in the experiment. Then X~B(3, 0.5) and the probability mass function of X given by . Therefore, the probability of obtaining at least 2 H’s is P(X ≥ 2) = P (X = 2 or X = 3) = P (X = 2) + P (X = 3) = ³C₂(0.5²)(0.5¹) + ³C₃(0.5³)(0.5⁰) = 0.375 + 0.125 = 0.5.

什么是正态分布？

Normal distribution is the continuous probability distribution defined by the probability density function, . The parameters denote the mean and the standard deviation of the population of interest. When the distribution is called the standard normal distribution.

这种分布称为正态分布，因为大多数自然现象遵循正态分布。例如，人类的智商是正态分布的。从图中可以看出，它是单峰的，关于平均值和钟形对称。平均值、模式和中位数是一致的。曲线下的面积对应于满足给定条件的人口比例。

The porti*** of population in the interval , , are approximately 68.2%, 95.6% and 99.8% respectively.

• 二项分布是一个离散的概率分布，而正态分布是一个连续的。
• 二项分布的概率质量函数是，而正态分布的概率密度函数是
• 在一定条件下，二项分布近似于正态分布，而不是相反。