information:extraction, stylistic:style_transfer
The moment of a convolution of function

The raw moment of a convolution h (t) = (f ∗ g) (t) = ∫ − ∞ ∞ f (τ) g (t − τ) d τ {\textstyle h(t)=(f*g)(t)=\int _{-\infty }^{\infty }f(\tau)g(t-\tau)\,d\tau } reads

where μ n {\displaystyle \mu _{n}} denotes the n {\displaystyle n} -th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain rule for differentiating a product.