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。