Search results
Results from the WOW.Com Content Network
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 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.
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
Even if the sample originates from a complex non-Gaussian distribution, it can be well-approximated because the CLT allows it to be simplified to a Gaussian distribution ("for a large number of observable samples, the sum of many random variables will have an approximately normal distribution").
2) random variable, then X 1 + X 2 is a normal (μ 1 + μ 2, σ 2 1 + σ 2 2) random variable. The sum of N chi-squared (1) random variables has a chi-squared distribution with N degrees of freedom. Other distributions are not closed under convolution, but their sum has a known distribution:
In its simplest form, it relates the expectation of a sum of randomly many finite-mean, independent and identically distributed random variables to the expected number of terms in the sum and the random variables' common expectation under the condition that the number of terms in the sum is independent of the summands.
As an example, the sum of two jointly normally distributed random variables, each with different means, will still have a normal distribution. On the other hand, a mixture density created as a mixture of two normal distributions with different means will have two peaks provided that the two means are far enough apart, showing that this ...
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.