二項分佈與正態分佈
隨機變量的概率分佈在統計學中佔有重要地位。在這些概率分佈中，二項分佈和正態分佈是現實生活中最常見的兩種。
什麼是二項分佈？
二項式分佈是與隨機變量X相對應的概率分佈，它是一系列獨立的是/否實驗的成功次數，每個實驗都有一個成功的概率p。從X的定義來看，它顯然是一個離散的隨機變量；因此，二項分佈也是離散的。
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.
什麼是正態分佈？
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.
這種分佈稱為正態分佈，因為大多數自然現象遵循正態分佈。例如，人類的智商是正態分佈的。從圖中可以看出，它是單峰的，關於平均值和鐘形對稱。平均值、模式和中位數是一致的。曲線下的面積對應於滿足給定條件的人口比例。
The porti*** of population in the interval , , are approximately 68.2%, 95.6% and 99.8% respectively.
- 二項分佈是一個離散的概率分佈，而正態分佈是一個連續的。
- 二項分佈的概率質量函數是，而正態分佈的概率密度函數是
- 在一定條件下，二項分佈近似於正態分佈，而不是相反。
