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In the event that the variables X and Y are jointly normally distributed random variables, then X + Y is still normally distributed (see Multivariate normal distribution) and the mean is the sum of the means. However, the variances are not additive due to the correlation. Indeed,
Random variables are assumed to have the following properties: complex constants are possible realizations of a random variable; the sum of two random variables is a random variable; the product of two random variables is a random variable; addition and multiplication of random variables are both commutative; and
In particular, the sum of the components X 1 + X 2 + · · · + X n is the riskiest if the joint probability distribution of the random vector (X 1, X 2, . . . , X n) is comonotonic. [2] Furthermore, the α-quantile of the sum equals the sum of the α-quantiles of its components, hence comonotonic random variables are quantile-additive.
When the random variables are also identically distributed , the Chernoff bound for the sum reduces to a simple rescaling of the single-variable Chernoff bound. That is, the Chernoff bound for the average of n iid variables is equivalent to the n th power of the Chernoff bound on a single variable (see Cramér's theorem ).
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. [1] The term 'random variable' in its mathematical definition refers to neither randomness nor variability [ 2 ] but instead is a mathematical function in which
A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
For reproductive models the weighted average of independent random variables with fixed μ and σ 2 and various values for w i is a member of the family of distributions with same μ and σ 2. The Tweedie exponential dispersion models are both additive and reproductive; we thus have the duality transformation Y ↦ Z = Y / σ 2 . {\displaystyle ...
Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free ...