Binomial
An algebraic expression with two terms
In probability theory and statistics, the binomial distribution is a discrete probability distribution that describes the probability of a certain number of successes out of a fixed number of independent trials, given the probability of success for each trial.
The distribution is commonly denoted by B(n, p), where n is the number of trials and p is the probability of success for each trial. The binomial distribution is characterized by two parameters: the number of trials and the probability of success for each trial.
The probability mass function (PMF) of a binomial random variable is given by:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials, k is the number of successes, p is the probability of success, and (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.
The mean (expected value) and variance of a binomial distribution are given by:
E(X) = np
Var(X) = np(1-p)
The binomial distribution is widely used in statistics and probability theory to model various real-world phenomena, such as the number of defective products in a manufacturing process, the number of people who choose a certain option in an election, and the number of accidents that occur in a given period of time.
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