The binomial distribution is the PMF of k successes given n independent events each with a probability p of success. Mathematically, when α = k + 1 and β = n − k + 1, the beta distribution and the binomial distribution are related by [clarification needed] a factor of n + 1:

在 概率论 和 统计学 中， 二项分布 （英語： binomial distribution）是一种 离散 概率分布，描述在进行 独立 随机试验 时，每次试验都有相同 概率 “成功”的情况下，获得成功的总次数。 掷硬币 十次出现五次正面的概率、产品合格率 时抽出一百件样本没有发现一件次品的概率等等，都可以由二项分布给出。 只有“成功”和“失败”两种 可能结果，每次重复时成功概率不变的独立随机试验称作 伯努利试验，例如上述的掷硬币出现正面或反面、对产品进行抽样检查时抽到正品或次品。 伯 …

Theorem 4.1 (Shorthand notation of a binomial distribution) (4.1) B (k; n, p) For the example of two heads in three coin tosses, we would write B (2; 3, 1 / 2). k is separated from the other parameters by a semicolon. This is because when we are talking about a distribution of values, we usually care about all values of k for a fixed n and p.

在概率论和统计学中，二项分布是n个独立的成功/失败试验中成功的次数的离散概率分布，其中每次试验的成功概率为p。 这样的单次成功/失败试验又称为 伯努利试验。 实际上，当n=1时，二项分布就是 伯努利分布。 [2] 一般地，如果随机变量 服从参数为 和 的二项分布，我们记为 或。 n次试验中正好得到k次成功的概率由概率质量函数给出： [2] 式中k=0，1，2，…，n, 是 二项式系数 （这就是二项分布名称的由来），又记为 或者。 该公式可以用以下方法理解：我们希望有k次 …

The shorthand X ∼ binomial(n,p)is used to indicate that the random variable X has the bino-mial distribution for positive integer parameter n and real parameter p satisfying 0 <p <1. The binomial distribution models the number of successes in n mutually independent Bernoulli trials, each with probability of success p.

The normal (Gaussian) and Lorentzian distributions are good examples of continuous distributions—the random variable can take on any value. Examples of discrete distributions include the Binomial, the Hypergeometric, and the Poisson. We will introduce the binomial today and then focus on the normal distribution.

If the conditions of the binomial setting are satisfied, then x, the number of successes, has a binomial distribution with parameters n and p; we express this distribution in shorthand as b(n, p). Now, we look at an example. As of August 2013, Jacqui Kalin was listed as having the top free throw percentage in Women’s NCAA basketball.

Working with the binomial CDF and pdfs can be tricky. Here are some tips on how to interpret word problems. We will consider a biased coin flip example where the number of trials is 40 and the probability that a heads (success) is 0.

A binomial distribution is used for discrete random variables with two possible outcomes (success or failure) and a fixed number of trials. It is characterized by the parameters n (number of trials) and p (probability of success).

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable [latex]X=[/latex] the number of successes obtained in the n independent trials. The mean, [latex]\mu[/latex], and variance, [latex]\sigma^{2}[/latex], for the binomial probability distribution are [latex]\mu=np[/latex] and [latex]\sigma^{2}=npq ...