Question

Which of the following expressions of the multivariate normal density function describes the shape of the density curve?

• A. \((2\pi)^{-p/2}|\Sigma|^{-1/2}\)
• B. \(|\Sigma|^{-1/2}\exp\left[-\frac12(x-\mu)'\Sigma^{-1}(x-\mu)\right]\)
• C. \((x-\mu)'\Sigma^{-1}(x-\mu)\)
• D. \(|\Sigma|^{-1/2}\)

Hint:

The normalizing constant only scales the height of the surface; the quadratic form inside the exponent fixes the elliptical contours and hence the shape.

Answer 1

The Correct Option is C

Step 1: Take logs of the density: $\ln f(x) = c - \dfrac12(x-\mu)'\Sigma^{-1}(x-\mu)$, where $c = -\dfrac{p}{2}\ln(2\pi) - \dfrac12\ln|\Sigma|$ collects every term that does not involve $x$.
Step 2: Since $c$ is a fixed number added everywhere, it only shifts $\ln f(x)$ up or down by the same amount for every $x$; it cannot change which $x$ values have higher or lower density relative to one another.
Step 3: The only $x$-dependent term is $-\dfrac12(x-\mu)'\Sigma^{-1}(x-\mu)$, so the level sets $\{x : (x-\mu)'\Sigma^{-1}(x-\mu) = k\}$ for a constant $k$ are exactly the contours of equal density, and these level sets are ellipsoids, the multivariate analogue of the univariate bell curve's shape.
Step 4: Therefore the quadratic form in the exponent, $(x-\mu)'\Sigma^{-1}(x-\mu)$, is the expression that governs the shape of the density surface.
\[ \boxed{(x-\mu)'\Sigma^{-1}(x-\mu)} \]