![]() Another famous early application of the normal distribution was by the British physicist James Clerk Maxwell, who in 1859 formulated his law of distribution of molecular velocities-later generalized as the Maxwell-Boltzmann distribution law.Or normal distribution. ![]() This study led Gauss to formulate his law of observational error and to advance the theory of the method of least squares approximation. The term “Gaussian distribution” refers to the German mathematician Carl Friedrich Gauss, who first developed a two-parameter exponential function in 1809 in connection with studies of astronomical observation errors. For further details see probability theory. Calculators have now all but eliminated the use of such tables. Although these areas can be determined with calculus, tables were generated in the 19th century for the special case of = 0 and σ = 1, known as the standard normal distribution, and these tables can be used for any normal distribution after the variables are suitably rescaled by subtracting their mean and dividing by their standard deviation, ( x − μ)/σ. Because the denominator (σ Square root of √ 2π), known as the normalizing coefficient, causes the total area enclosed by the graph to be exactly equal to unity, probabilities can be obtained directly from the corresponding area-i.e., an area of 0.5 corresponds to a probability of 0.5. The probability of a random variable falling within any given range of values is equal to the proportion of the area enclosed under the function’s graph between the given values and above the x-axis. In this exponential function e is the constant 2.71828…, is the mean, and σ is the standard deviation. The normal distribution is produced by the normal density function, p( x) = e −( x − μ) 2/2σ 2/σ Square root of √ 2π.
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