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Probability Theory

The origin of probability theory lies in physical observations associated with games of chance. Many actions have outcomes which are largely unpredictable in advance-tossing a coin and throwing a dart are simple examples. Probability theory is about such actions and their consequences. The mathematical theory starts with the idea of an experiment (or trial), being a course of action whose consequence is not predetermined; this experiment is reformulated as a mathematical object called a probability space. In broad terms the probability space corresponding to a given experiment comprises three items:
  • The set of all possible outcomes of the experiment.
  • A list of all the events which may possibly occur as consequences of the experiment.
  • An assessment of the likelihoods of these events. 

Given any experiment involving chance, there is a corresponding probability space and the study of such spaces is called probability theory. 

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Gaussian and Poisson Distributions

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The Poisson Distribution:
The binomial distribution describes cases where particular outcomes occur in a certain number of trials, n. The Poisson distribution describes cases where there are still particular outcomes but no idea of the number of trials; instead these are sharp events occurring in a continuum.
The Gaussian Distribution:
The Gaussian or normal is the most well known and useful of all distributions. It is a bell-shaped curve centered on and symmetric about x = μ. The width is controlled by the parameter sigma which is also the standard deviation of the distribution.

Error Analysis

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For some reason the word 'error' is unpopular in our context. One reason might be destruct of the concept of an ideal result of measurement-without which the terms 'accuracy' and 'error' lack meaning. But, as already indicated, if there is no such thing as an ideal result then the experimentalist has to find an alternative logic when using the concepts of probability. Another potential reason is that the word 'error' might be thought to imply that a mistake has been made and the measurement scientist rightly wishes to avoid giving that impression. But perhaps the principal cause of the unpopularity of the word 'error' in measurement science is a perception that the techniques of so called 'error analysis' are inadequate. The techniques of the subject that was known by that name in the middle of the twentieth century do indeed seem limited.

Significant Figures

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Numbers obtained from a measurement are approximate values. There is always some uncertainty due to the limitations of the measuring devices used and the skill of the individual making the measurement. The figures used to report a result should reflect the precision of the test and the sensitivity of the measuring device that produced the value. To express this precision, the number should contain all the digits that are known plus one digit that is estimated. These are the significant figures. The term significant refers to the number of digits that are meaningful with regard to the accuracy of the value.

Least Square Fitting

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Data fitting is the tool used by experimentalists to verify a theoretical prediction, many more points than the minimum are measured in order to minimize the effects of random errors generated in the acquisition of the data. But this over determination in the system parameters faces us with the dilemma of what confidence level one gives to the accuracy of specific data points and which data points to accept or reject.

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