finite

Examples of finite in the following topics:

• Counting Rules and Techniques

  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
  • A bijective proof is a proof technique that finds a bijective function $f: A \rightarrow B$ between two finite sets $A$ and $B$, which proves that they have the same number of elements, $|A| = |B|$.
  • In this technique, a finite set $X$ is described from two perspectives, leading to two distinct expressions for the size of the set.

• The Hypergeometric Random Variable

  • The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes.
  • The hypergeometric distribution applies to sampling without replacement from a finite population whose elements can be classified into two mutually exclusive categories like pass/fail, male/female or employed/unemployed.

• The Role of the Model

  • The probability distributions describing the data-generation process are assumed to be fully described by a family of probability distributions involving only a finite number of unknown parameters.
  • For example, one may assume that a population distribution has a finite mean.

• Continuous Sampling Distributions

  • This distribution was discrete, since there were a finite number of possible observations.

• Categorical Data and the Multinomial Experiment

  • In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number $k$ of possible outcomes, with probabilities $p_1, \cdots , p_k$ (so that $p_i \geq 0$ for $i = 1, \cdots, k$ and the sum is $1$), and there are $n$ independent trials.

• Summation Notation

  • For finite sequences of such elements, summation always produces a well-defined sum.
  • Addition is also commutative, so changing the order of the terms of a finite sequence does not change its sum.

• Standard Error

  • When the sampling fraction is large (approximately at 5% or more), the estimate of the error must be corrected by multiplying by a "finite population correction" to account for the added precision gained by sampling close to a larger percentage of the population.

• Sampling Distribution of the Mean

  • Given a population with a finite mean $\mu$ and a finite non-zero variance $\sigma$, the sampling distribution of the mean approaches a normal distribution with a mean of $\mu$ and a variance of $\frac{\sigma^2}{N}$ as N, the sample size, increases.

• Least-Squares Regression

  • Under these conditions, the method of OLS provides minimum-variance, mean-unbiased estimation when the errors have finite variances.

• Homogeneity and Heterogeneity

  • In technical terms, a data set is homoscedastic if all random variables in the sequence have the same finite variance.