 # Trials

## Trials Topics

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### Sample

A sample is a subset of a population that is obtained through some process, possibly random selection or selection based on a certain set of criteria, for the purposes of investigating the properties of the underlying parent population. In particular, statistical quantities determined directly from the sample (such as sample central moments, sample raw moments, sample mean, sample variance, etc.) can be used as estimators for the corresponding properties of the underlying distribution.The process of obtaining a sample is known as sampling, and the number of members in a sample is called the sample size.

### Lexis trials

sets of trials each, with the probability of success constant in each set.where is the variance of .

### Lexis ratio

where is the variance in a set of Lexis trials and is the variance assuming Bernoulli trials. If , the trials are said to be subnormal, and if , the trials are said to be supernormal.

### Supernormal

Trials for which the Lexis ratiosatisfies , where is the variance in a set of Lexis trials and is the variance assuming Bernoulli trials.

### Poisson trials

A number of trials in which the probability of success varies from trial to trial. Let be the number of successes, then(1)where is the variance of and . Uspensky has shown that(2)where(3)(4)(5)(6)and . The probability that the number of successes is at least is given by(7)Uspensky gives the true probability that there are at least successes in trials as(8)where(9)(10)

### Experiment

An experiment is defined (Papoulis 1984, p. 30) as a mathematical object consisting of the following elements. 1. A set (the probability space) of elements. 2. A Borel field consisting of certain subsets of called events. 3. A number satisfying the probability axioms, called the probability, that is assigned to every event .

### Sample proportion

Let there be successes out of Bernoulli trials. The sample proportion is the fraction of samples which were successes, so(1)For large , has an approximately normal distribution. Let RE be the relative error and SE the standard error, then(2)(3)(4)where CI is the confidence interval and is the erf function. The number of tries needed to determine with relative error RE and confidence interval CI is(5)