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Otherwise, summation is denoted by using Σ notation, where is an enlarged capital Greek letter sigma. For example, the sum of the first n natural numbers can be denoted as = For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For ...
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]
Using sigma summation notation the sum of the first m terms of the series can be expressed as = (). The infinite series diverges , meaning that its sequence of partial sums , (1, −1, 2, −2, 3, ...) , does not tend towards any finite limit .
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [8] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
In the social sciences, a result may be considered statistically significant if its confidence level is of the order of a two-sigma effect (95%), while in particle physics and astrophysics, there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a discovery.
The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges. [35]
When z is 1, the function is called the sigma function or sum-of-divisors function, [1] [3] and the subscript is often omitted, so σ(n) is the same as σ 1 (n) (OEIS: A000203). The aliquot sum s ( n ) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS : A001065 ), and equals σ 1 ( n ) − n ; the aliquot ...
Using matrix notation, the sum of squared residuals is given by S ( β ) = ( y − X β ) T ( y − X β ) . {\displaystyle S(\beta )=(y-X\beta )^{T}(y-X\beta ).} Since this is a quadratic expression, the vector which gives the global minimum may be found via matrix calculus by differentiating with respect to the vector β {\displaystyle \beta ...