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In measure-theoretic probability theory, the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. [5] The likelihood function is this density interpreted as a function of the parameter, rather than the random variable. [ 6 ]
In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, [1] i.e. by means not of a theoretical sample space but of an actual experiment.
The cumulative frequency is the total of the absolute frequencies of all events at or below a certain point in an ordered list of events. [1]: 17–19 The relative frequency (or empirical probability) of an event is the absolute frequency normalized by the total number of events:
The relative entropy was introduced by Solomon Kullback and Richard Leibler in Kullback & Leibler (1951) as "the mean information for discrimination between and per observation from ", [6] where one is comparing two probability measures ,, and , are the hypotheses that one is selecting from measure , (respectively).
As the number of trials increases, the relative frequencies approach their respective theoretical probabilities, demonstrating the Law of Large Numbers. For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1 ⁄ 2.
The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio , relative risk measures the association between the exposure and the outcome.
The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin.
The relative likelihood of θ is ... the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is ...
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