# 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 .XφX (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 .XMX (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:

- If are independent, then it is for everyone and for everyone
*for which the mgf exist*.X1,…, XnφX1 + ⋯ + Xn (t) = φX1 (t) ⋯ φXn (t)tMX1 + ⋯ + Xn (t) = MX1 (t) ⋯ MXn (t)t - 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) t*0*XY - 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

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