binomial distribution

简明释义

1. 二项分布; 2. 二项式分布;

英英释义

例句

1.The probability of getting heads in a series of coin flips can be calculated with binomial distribution (二项分布).

在一系列抛硬币中得到正面的概率可以通过二项分布（二项分布）来计算。

2.In genetics, the inheritance of traits can be predicted using binomial distribution (二项分布).

在遗传学中，特征的遗传可以使用二项分布（二项分布）进行预测。

3.In clinical trials, the success rate of a treatment can be analyzed using binomial distribution (二项分布).

在临床试验中，治疗的成功率可以使用二项分布（二项分布）进行分析。

4.A survey asking whether people prefer tea or coffee can use binomial distribution (二项分布) to analyze preferences.

一项调查询问人们是否喜欢茶或咖啡，可以使用二项分布（二项分布）来分析偏好。

5.In a quality control process, the number of defective items can be modeled using binomial distribution (二项分布).

在质量控制过程中，缺陷产品的数量可以用二项分布（二项分布）来建模。

作文

The concept of binomial distribution is a fundamental aspect of probability theory that is widely used in statistics. It describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Understanding binomial distribution is crucial for various fields, including economics, psychology, and health sciences, where researchers often need to analyze the outcomes of experiments or surveys. To illustrate the significance of binomial distribution, let’s consider a simple example. Imagine a scenario where a fair coin is flipped ten times. We want to know the probability of getting exactly six heads. In this case, each flip of the coin represents a Bernoulli trial, where there are only two possible outcomes: heads (success) or tails (failure). The probability of getting heads on any single flip of a fair coin is 0.5. Using the formula for binomial distribution, we can calculate the probability of getting exactly six heads in ten flips. The formula is given by:P(X = k) = (n choose k) * p^k * (1-p)^(n-k)Where:- P(X = k) is the probability of getting exactly k successes in n trials.- (n choose k) is the binomial coefficient, which counts the number of ways to choose k successes from n trials.- p is the probability of success on an individual trial.- (1-p) is the probability of failure.In our example, n is 10 (the number of flips), k is 6 (the number of heads we want), and p is 0.5. Plugging these values into the formula gives us:P(X = 6) = (10 choose 6) * (0.5)^6 * (0.5)^(10-6)Calculating this yields the probability of obtaining exactly six heads in ten flips of the coin. This process showcases how binomial distribution allows us to quantify uncertainty and make predictions based on known probabilities. Moreover, the application of binomial distribution extends beyond simple coin tosses. In real-world scenarios, it can be applied to various situations such as quality control in manufacturing, where a company may want to determine the likelihood of a certain number of defective items in a batch of products. Similarly, in medical research, binomial distribution can help analyze the effectiveness of a new drug by determining how many patients out of a group respond positively to the treatment. One important aspect of binomial distribution is its relation to the normal distribution. According to the Central Limit Theorem, as the number of trials increases, the binomial distribution approaches a normal distribution. This property allows statisticians to use normal approximation methods for large sample sizes, simplifying calculations and analyses. In conclusion, understanding binomial distribution is essential for anyone involved in data analysis and decision-making processes. Its applications are vast, ranging from academic research to practical business scenarios. By mastering the principles of binomial distribution, individuals can better interpret data, assess risks, and make informed decisions based on statistical evidence.

二项分布的概念是概率论的一个基本方面，在统计学中广泛应用。它描述了在固定数量的独立伯努利试验中成功的次数，每次试验都有相同的成功概率。理解二项分布对于经济学、心理学和健康科学等各个领域至关重要，因为研究人员通常需要分析实验或调查的结果。为了说明二项分布的重要性，让我们考虑一个简单的例子。想象一下，一个公平的硬币被抛掷十次。我们想知道得到六个正面的概率。在这种情况下，硬币的每次抛掷都代表一次伯努利试验，只有两种可能的结果：正面（成功）或反面（失败）。在任何一次公平硬币的抛掷中，得到正面的概率是0.5。使用二项分布的公式，我们可以计算在十次抛掷中得到恰好六个正面的概率。公式如下：P(X = k) = (n choose k) * p^k * (1-p)^(n-k)其中：- P(X = k) 是在 n 次试验中恰好获得 k 次成功的概率。- (n choose k) 是二项系数，计算从 n 次试验中选择 k 次成功的方式数。- p 是单次试验成功的概率。- (1-p) 是失败的概率。在我们的例子中，n 为 10（抛掷次数），k 为 6（我们想要的正面次数），p 为 0.5。将这些值代入公式，可以得出在十次抛掷中恰好得到六个正面的概率。这个过程展示了二项分布如何让我们量化不确定性，并根据已知概率做出预测。此外，二项分布的应用超越了简单的硬币投掷。在现实世界场景中，它可以应用于各种情况，例如制造中的质量控制，公司可能希望确定一批产品中有多少个缺陷产品。同样，在医学研究中，二项分布可以帮助分析新药的有效性，确定一组患者中有多少人对治疗产生积极反应。二项分布的一个重要方面是它与正态分布的关系。根据中心极限定理，随着试验次数的增加，二项分布趋近于正态分布。这一特性使统计学家能够在样本量较大时使用正态近似方法，从而简化计算和分析。总之，理解二项分布对任何参与数据分析和决策过程的人来说都是必不可少的。它的应用范围广泛，从学术研究到实际商业场景。通过掌握二项分布的原理，个人可以更好地解释数据、评估风险，并根据统计证据做出明智的决策。