Lecture V Probability theory

Lecture questions Classical definition of probability Frequency probability Discrete variable and probability distribution Continuous variable and probability density function

Probability theory Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes (random processes), and events. The probability theory is a mathematical foundation for statistics. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state.

Basic Concepts Random Experiment. An experiment is said to be a random experiment, if it's out-come can't be predicted with certainty. Sample Space. The set of all possible out-comes of an experiment is called the sample space. The elements of the sample space are called sample point or event point. Event. Every subset of a sample space is an event. Simple event. An event, consisting of a single sample point is called a simple event. Compound event. A subset of the sample space, which has more than one element is called a mixed event. Exhaustive events. When every possible outcome of an experiment is considered.

Classical definition of probability The probability of an event is the ratio of the number of cases m favorable for the event to the number of total outcomes n(s) possible in an equiprobable sample space. Note: This definition is not true, if (a) The events are not equiprobable. (b) The possible outcomes are infinite.

Statistical probability (frequency probability) Frequency (absolute frequency) is the number of times an event actually occurs in any experiment. The relative frequency of an event is the ratio of the absolute frequency of a particular successful event, with the total number of events. Statistical probability (frequency probability) defines as the limit of relative frequency of an event in a large number of trials.

Discrete and continuous variables A random variable (or stochastic variable) is a way of assigning a value (often a real number to each possible outcome of a random event. A discrete random variable maps events to values of a countable set (e.g., the integers, with each value in the range having probability greater than or equal to zero. A continuous random variable maps events to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive.

Probability distribution A probability distribution identifies the probability of each value of a random variable when the variable is discrete and the probability of the value falling within a particular interval when the variable is continuous. The probability distribution of a continuous random variable is completely described by probability density function.

Moments The probability distribution of a random variable is often characterized by a small number of parameters, which also have a practical interpretation. They are The expected value The variance Standard deviation

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