Search results
Results from the WOW.Com Content Network
The Brahmagupta–Fibonacci identity states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, given n = a 2 + b 2 = c 2 + d 2 {\displaystyle n=a^{2}+b^{2}=c^{2}+d^{2}} we find n {\displaystyle n} as a product of sums of two squares.
For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares.
The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2. The prime decomposition of the number 3430 is 2 · 5 · 7 3. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.
More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is, a 2 − b 2 = ( a + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(a-b)} This is the difference-of-squares formula , which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 50 2 − 3 2 ...
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
The product of a finite number of characteristic functions is also a characteristic function. The same holds for an infinite product provided that it converges to a function continuous at the origin. If φ is a characteristic function and α is a real number, then φ ¯ {\displaystyle {\bar {\varphi }}} , Re( φ ), | φ | 2 , and φ ( αt ) are ...