| ### Appendix C - Jitter Calculations
Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks
Russell D. Reed and Robert J. Marks II
Copyright © 1999 Massachusetts Institute of Technology

## C.3 Jitter: CDF-PDF Convolution in One Dimension

The following demonstrates that the convolution of Gaussian PDF with variance σ21 and a Gaussian CDF with variance σ22 results in a Gaussian CDF with variance σ23 = σ21 + σ22. All the functions are one-dimensional.

Consider two independent random variables X1 and X2 with PDFs f1 and f2 and CDF's F1 and F2. The random variable Y = X1 + X2 has the PDF f1 * f2 and consequently its CDF is F1 * f2 = f1 * F2. Let X1 and X2 be zero mean Gaussian, X1 ~ N(0, σ21) and X2 ~ N(0, σ22), then, clearly, Y ~ N(0, σ21 + σ22) has a Gaussian PDF with variance σ23 = σ21 + σ22. Because Y has the CDF f1 * F2, f1 * F2 is a Gaussian CDF with zero mean and variance σ23 = σ21 + σ22.