What is the MGF of the normal distribution

When is the moment generating function preferable to the characteristic function?


Let be a probability space and be a random vector. Let be the distribution of, a Borel measure for. (Ω, F, P) X: Ω → RnPX = X ∗ PXRn

  • The characteristic function of is the function defined for (the random variable is therefore in for all). This is the Fourier transform of .X
    φX (t) = E [eit⋅X] = ∫Ωeit⋅XdP,
    t∈Rneit⋅XL1 (P) tPX
  • The Torque generation function ( mgf ) from is the function defined for all for which the above integral exists . This is the Laplace transform of .X
    MX (t) = E [et⋅X] = ∫Ωet⋅XdP,
    t∈RnPX

We can already see that the characteristic function is everywhere in, but the mgf has a domain that depends on, and this domain can only (this happens for example for a Cauchy-distributed random variable) .RnX {0}

Nevertheless, characteristic functions and mgf have many properties in common, for example:

  1. If are independent, then it is for everyone and for everyone for which the mgf exist .X1,…, Xn
    φX1 + ⋯ + Xn (t) = φX1 (t) ⋯ φXn (t)
    t
    MX1 + ⋯ + Xn (t) = MX1 (t) ⋯ MXn (t)
    t
  2. Two random vectors and have the same distribution if and only for all. The mgf analog of this result is that if for everyone in a neighborhood of , and have the same distribution: XYφX (t) = φY (t) tMX (t) = MY (t) t0XY
  3. Characteristic functions and mgFs of common distributions often have similar shapes. For example, if (dimensional normals with mean and covariance matrix), then and X∼Nn (μ, Σ) nμΣ