BASIC CONCEPTS IN PROBABILITY THEORY

Probability theory deals with the expected frequencies of various events in random sampling. The set of events considered in the sampling is called the sample space of the given problem, and may be discrete (like heads and tails in coin tossing) or continuous (like the set of values on the real number line). 

For a discrete sample space of mutually exclusive events Ei, one defines a corresponding set of event probabilities pi = p Ei), which may be given or may be estimated from observations. The Pi have the properties 

For a continuous sample space, one associates the probabilities not with points, but with differential regions of the space. Then the probability of obtaining a sample value in the interval (xq /2,xo + Axj2) is expressed 

The probability of obtaining a sample value in the region x is then 

For later discussions, it is useful to extend the concepts of probability density and distribution to sample spaces of discrete points x. Equations (4.2-4) to (4.2-6) hold directly here also, if we define the density function for this case as a set of spikes of zero width (impulse functions), with included probability p(x ) for each Xj. The resulting distribution function F f) is the combined probability of the sample points Xi f. 

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