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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.
In probability theory, 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 ...
These Gaussians are plotted in the accompanying figure. The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances: = +. The product of two Gaussian probability density functions (PDFs), though, is not in general a Gaussian PDF.
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y , the distribution of the random variable Z that is formed as the product Z = X Y {\displaystyle Z=XY} is a product distribution .
The term convolution refers to both the resulting function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function.
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]
Depending on the values of the parameters, the distribution may vary in shape from almost normal to almost exponential. The parameters of the distribution can be estimated from the sample data with the method of moments as follows: [4] [5] = +, = +,
For medium size samples (<), the parameters of the asymptotic distribution of the kurtosis statistic are modified [37] For small sample tests (<) empirical critical values are used. Tables of critical values for both statistics are given by Rencher [38] for k = 2, 3, 4.