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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 density of the sum of two independent real-valued random variables equals the convolution of the density functions of the original variables. Thus, the density of the sum of m+n terms of a sequence of independent identically distributed variables equals the convolution of the densities of the sums of m terms and of n term. In particular ...
In particular, when two or more random variables are statistically independent, the n th-order cumulant of their sum is equal to the sum of their n th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property.
Let a random variable ξ be normally distributed and admit a decomposition as a sum ξ=ξ 1 +ξ 2 of two independent random variables. Then the summands ξ 1 and ξ 2 are normally distributed as well. A proof of Cramér's decomposition theorem uses the theory of entire functions.
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.
The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal. The normal-exponential-gamma distribution
A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.
Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as: V a r [ f ( X ) ] ≈ ∑ n = 1 n m a x ( σ n n ! ( d n f d X n ) X = μ ) 2 V a r [ Z n ] + ∑ n = 1 n m a x ∑ m ≠ n σ n + m n ! m !