Convolution of probability distributions - Wikipedia
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Abstraction: Summing independent random variables via distribution convolution
Key points:
- Distribution of sum of independent random variables equals convolution of their individual distributions
- For discrete variables: PMF of sum is convolution of individual PMFs; for continuous: PDF of sum is convolution of PDFs
- CDF of sum Z = X + Y: F_Z(z) = integral of F_X(z - y) dF_Y(y) when independent
- Characteristic functions simplify computation: characteristic function of sum equals product of individual characteristic functions
- Classic example: sum of two independent Bernoulli(p) variables is Binomial(2, p), proven via Pascal's rule
- Many well-known distributions have closed-form convolutions listed in Wikipedia's companion table
Connections: Wikipedia · Probability Theory · Statistics
Source: http://en.wikipedia.org/wiki/Convolution_of_probability_distributions