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.