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... so any square is congruent to 0 or 1 mod 4, hence p 3 mod 4 cannot be a sum of two squares. The remaining case of p = 2 needs to be treated separately but is trivial. In fact, we can derive a complete characterization of integers that are sums of two squares: Corollary 2.10 (Fermat's two square th ...
... so any square is congruent to 0 or 1 mod 4, hence p 3 mod 4 cannot be a sum of two squares. The remaining case of p = 2 needs to be treated separately but is trivial. In fact, we can derive a complete characterization of integers that are sums of two squares: Corollary 2.10 (Fermat's two square th ...
for all - WorkBank247.com
... Since an is any Cauchy sequence of real numbers, so by theorem we know that every Cauchy sequence of real numbers converges to a limit. Since bn is any decreasing sequence of positive real numbers, so it is bounded. Then the sequence is convergent. By a limit theorem, we know the limit of the diffe ...
... Since an is any Cauchy sequence of real numbers, so by theorem we know that every Cauchy sequence of real numbers converges to a limit. Since bn is any decreasing sequence of positive real numbers, so it is bounded. Then the sequence is convergent. By a limit theorem, we know the limit of the diffe ...
2. Primes Primes. • A natural number greater than 1 is prime if it
... that is, the sets of primes are equal but they may not be in the same order, so pi may or may not equal qi . Moreover, if pi = qj , then ri = tj . In other words, if we express the same natural number as a product of powers of distinct primes, then the expressions are identical except for the orderi ...
... that is, the sets of primes are equal but they may not be in the same order, so pi may or may not equal qi . Moreover, if pi = qj , then ri = tj . In other words, if we express the same natural number as a product of powers of distinct primes, then the expressions are identical except for the orderi ...
QUIZ #1
... These slides were created by Tom Hettmansperger and in some cases modified by David Hunter ...
... These slides were created by Tom Hettmansperger and in some cases modified by David Hunter ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.