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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.
This is called the addition law of probability, or the sum rule. That is, the probability that an event in A or B will happen is the sum of the probability of an event in A and the probability of an event in B, minus the probability of an event that is in both A and B. The proof of this is as follows: Firstly,
5+0=5 illustrated with collections of dots. In combinatorics, the addition principle [1] [2] or rule of sum [3] [4] is a basic counting principle.Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways of doing another thing and we can not do both at the same time, then there are + ways to choose one of the actions.
The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things).
Sum rule may refer to: Sum rule in differentiation, Differentiation rules #Differentiation is linear; Sum rule in integration, see Integral #Properties; Addition principle, a counting principle in combinatorics; In probability theory, an implication of the additivity axiom, see Probability axioms #Further consequences; Sum rule in quantum mechanics
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
A Binomial distributed random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables X ~ B(n, p) and Y ~ B(m, p) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z = X + Y ~ B(n + m, p). This can also be proven ...
The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory.