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For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself.
The summation can be interpreted as a weighted average, and consequently the marginal probability, (), is sometimes called "average probability"; [2] "overall probability" is sometimes used in less formal writings. [3] The law of total probability can also be stated for conditional probabilities:
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]
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...
According to the summation formula in the case of random variables with countably many outcomes, one has [] = = = + + + + = + + + +. It is natural to say that the expected value equals +∞ . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. [ 19 ]
In probability theory, Wald's equation, Wald's identity [1] or Wald's lemma [2] is an important identity that simplifies the calculation of the expected value of the sum of a random number of random quantities.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
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.