Mathematical statistics

                           MATHEMATICAL STATISTICS

Mathematical statistics is the application of mathematics to statistics, which was originally conceived as the science of the state — the collection and analysis of facts about a country: its economy, land, military, population, and so forth. Mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory.

Probability distributions

A probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment,survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution can either be univariate or multivariate. A univariate distribution gives the probabilities of a singlerandom variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, thehypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

Special distributions

• Normal distribution (Gaussian distribution), the most common continuous distribution
• Bernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)
• Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) given a fixed total number of independent occurrences
• Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
• Geometric distribution, for binomial-type observations but where the quantity of interest is the number of failures before the first success; a special c*Discrete uniform distribution, for a finite set of values (e.g. the outcome of a fair die)
• Continuous uniform distribution, for continuously distributed values
• Poisson distribution, for the number of occurrences of a Poisson-type event in a given period of time
• Exponential distribution, for the time before the next Poisson-type event occurs
• Gamma distribution, for the time before the next k Poisson-type events occur
• Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g. for inference regarding the sample variance of normally distributed samples (see chi-squared test)
• Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable; useful for inference regarding the mean of normally distributed samples with unknown variance (seeStudent's t-test)
• Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoulli distribution andbinomial distribution

geometry

                                                      GEOMETRY
Geometry  is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

• Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study ofpoints, lines, planes, angles, triangles, congruence, similarity, solid figures,circles, and analytic geometry.^[7] Euclidean geometry also has applications incomputer science, crystallography, and various branches of modern mathematics.
• Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics, including in general relativity.
• Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, such as connectedness andcompactness.
• Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.
• Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in many areas, including cryptography and string theory.
• Discrete geometry is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. It shares many methods and principles with combinatorics.

                        Vedic Mathematics 

Vedic Mathematics is a book written by the Indian Hindu priest Bharati Krishna Tirthaji and first published in 1965. It contains a list of mental calculation techniques claimed to be based on the Vedas. The mental calculation system mentioned in the book is also known by the same name or as "Vedic Maths". Its characterization as "Vedic" mathematics has been criticized by academics, who have also opposed its inclusion in the Indian school curriculum.

Publication history[edit]

Although the book was first published in 1965, Tirthaji had been propagating the techniques since much earlier, through lectures and classes.^[1]He wrote the book in 1957.^[2]:10 It was published in 1965, five years after his death as 367 pages in forty chapters. Reprints were made in 1975 and 1978 with fewer typographical errors.^[3] Several reprints have been made since the 1990s.^[2]:6

Use in schools[edit]

The book was previously included in the school syllabus of Madhya Pradesh and Uttar Pradesh.^[2]:6 Some schools and organizations run by Hindu nationalist groups, including those outside India, have also included Tirthaji's techniques in their curriculum. The Hindu nationalists have also made several attempts to have Tirthaji's "Vedic mathematics" system included in the Indian school curriculum via the NCERT books.

A number of academics and mathematicians have opposed these attempts on the basis that the techniques mentioned in the book are simply arithmetic tricks, and not mathematics. They also pointed out that the term "Vedic" mathematics is incorrect, and there are other texts that can be used to teach a correct account of the Indian mathematics during the Vedic period. They also criticized the move as a saffronization attempt to promote religious majoritarianism.^[9][10]

Dani points out that while Tirthaji's system could be used as a teaching aid, there was a need to prevent the use of "public money and energy on its propagation, beyond the limited extent". He pointed out that the authentic Vedic studies had been neglected in India even as Tirthaji's system received support from several Government and private agencies.^[1]

Proponents of Vedic Mathematics however argue that the methods are not merely mathematical tricks and that there is an underlying psychology because the aphorisms describe personal approaches to problem-solving. As pedagogic tools, the methods are useful because they invite students to deal with strategies.^[11]

Hipparchus:Father of trignometry

                                             Father of Trignometry

Life and work

Relatively little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. Most of what is known about Hipparchus comes from Strabo's Geography and Pliny's Natural History in the 1st century; Ptolemy's 2nd-century Almagest; and additional references to him in the 4th century by Pappus of Alexandria and Theon of Alexandria in their commentaries on the Almagest.^[4]

There is a strong tradition that Hipparchus was born in Nicaea (Greek Νίκαια), in the ancient district of Bithynia (modern-day Iznik in province Bursa), in what today is the country Turkey.

The exact dates of his life are not known, but Ptolemy attributes to him astronomical observations in the period from 147–127 bc, and some of these are stated as made in Rhodes; earlier observations since 162 bc might also have been made by him. His birth date (c. 190 bc) was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 bc because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life.

Geometry, trigonometry, and other mathematical techniques

Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals the radius times twice the sine of half of the angle, i.e.:

chord(A) = r(2 sin(A/2)).

He described the chord table in a work, now lost, called Tōn en kuklō_i eutheiōn (Of Lines Inside a Circle) by Theon of Alexandria in his 4th-century commentary on the Almagest I.10; some claim his table may have survived in astronomical treatises in India, for instance the Surya Siddhanta. Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques.^[19]

For his chord table Hipparchus must have used a better approximation for π than the one from Archimedes of between 3 + 1/7 and 3 + 10/71; perhaps he had the one later used by Ptolemy: 3;8:30 (sexagesimal) (Almagest VI.7); but it is not known if he computed an improved value himself.

But some scholars do not believe Arayabhatta's Sin table has anything to do with Hipparchus's chord table which does not exist today. Some scholars do not agree with this hypothesis that Hipparchus constructed a chord table. Bo. C Klintberg states "With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer's reconstructions with a 3600' radius – i.e. the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees' – generates Hipparchan-like ratios similar to those produced by a 3438' radius. It is therefore possible that the radius of Hipparchus's chord table was 3600', and that the Indians independently constructed their 3438'-based sine table." ^[20]

Hipparchus could construct his chord table using the Pythagorean theorem and a theorem known to Archimedes. He also might have developed and used the theorem in plane geometry called Ptolemy's theorem, because it was proved by Ptolemy in his Almagest (I.10) (later elaborated on by Carnot).

Hipparchus was the first to show that the stereographic projection is conformal, and that it transforms circles on the sphere that do not pass through the center of projection to circles on the plane. This was the basis for the astrolabe.

Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers.

There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text of it is that of Menelaus of Alexandria in the 1st century, who on that basis is now commonly credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things like the rising and setting points of the ecliptic, or to take account of the lunar parallax. Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. He might have used spherical trigonometry.

Aubrey Diller has shown that the clima calculations which Strabo preserved from Hipparchus were performed by spherical trigonometry with the sole accurate obliquity known to have been used by ancient astronomers, 23°40'. All thirteen clima figures agree with Diller's proposal.^[21] Further confirming his contention is the finding that the big errors in Hipparchus's longitude of Regulus and both longitudes of Spica agree to a few minutes in all three instances with a theory that he took the wrong sign for his correction for parallax when using eclipses for determining stars' positions.^[22]