The Gaussian Distribution from Scratch

2. The moment-generating function is MX(t) := E[exp(t >X)] = exp(t>μ+ 1 2 t >Σt) for all t ∈ R. 3. X = μ+ ΣZ where Z ∼ N (0d, Id×d). 4. a>X ∼ N (a>μ, a>Σa) for all nonzero a ∈ R. If any holds, we say X ∈ R is normally distributed with parameters (μ,Σ). Note that 3 and 4 just reduce general normality to simpler forms of (1) (standard and univariate). These alternative definitions are useful in different contexts, for instance