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Discrete & Continuous Probability Distributions, Stats Assignment Help
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>> Discrete & Continuous Probability Distributions
The word probability is derived from the Latin word probabilitias which means probity. It is used for describing an attitude of mind for some proposition of whose truth or result we are not certain about. Higher probability for any event means there is more certainty to the occurring of event. Hence, the use of probability in applied sense is meant to measure the likeliness of the occurring of an event.
The probability theory is that branch of mathematics which deals with random phenomena. The probability theory centers on random variables, stochastic events and processes. Probability theory is essential in human activities which involve quantitative analysis for larger sets of data. Physics is the study of probabilistic nature of physical phenomena’s occurring at the atomic scales and described by quantum mechanics.
Probability of an event
- an event is termed as a given set of outcomes for which the probability is assigned. Since the sample space is finite, any subset in the sample space can be said to be an event. For example, in an assembly of 52 playing cards in a deck having no jokers, the sample space is a 52-element set and if we draw a single card from the deck then withdrawing of each card is a possible outcome. But the event is a subset of the sample space and may include a singleton set or elementary event, an empty set, a possible event and even the sample space.
- It is the probability of any event in a particular set of given circumstances. In more formal terms, a conditional probability can be termed as the occurring of event A when the sample space is limited to the event B. The conditional probability is denoted by P (A/B) and is read as ‘the probability of A, given b’.
Discrete and continuous probability distributions-
the probability mass, probability distribution, or probability density is a function which describes the probability of any random variable by taking and assuming certain values. In discrete probability distributions, this random variable is a discrete variable. For example, flipping a coin two times gives four possible outcomes, HH, HT, TH and TT (H for head and T for tail).In case the random variable X represents the number of heads which result form the experiment, then it can only take the values of 0, 1 or 2 and hence is discrete random variable. In continuous probability distribution, the random variable is a continuous variable. This continuous variable is different from the discrete variable as the probability that a continuous variable might assume any particular value is zero. Therefore a continuous probability distribution cannot be expressed in a tabular form but is described by using an equation or formulas.
History of probability theory
The roots of the mathematical theory of probability can be found in the attempts made for analyzing the game of chance by the mathematician Gerolamo Cardano in the early sixteenth century. The contribution of Pierre de Fermat and Blasé Pascal in the problem of points had some important repercussions on the theory of probability. A full book on the subject was published by Christian Huygens in the year 1657. In the beginning, probability theory included only the discrete events but the analytical considerations led to the incorporation of the continuous variables in the probability theory. The modern probability theory has its foundation laid by Andrei Nikolayevich Kolmogorov, who presented his axiom system of probability in the year 1933 by combining the notion of sample space (given by Richard von Mises) and the measure theory. Later Bruno de Fineti combined the finite additivity in place of the countable additivity with the theory, and gave another alternative to the probability theory.
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