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: The multivariate normal density for a \(p\)-variate vector \(x\) with mean \(\mu\) and covariance \(\Sigma\) is \[f(x) = (2\pi)^{-p/2}|\Sigma|^{-1/2}\exp\left[-\frac12(x-\mu)'\Sigma^{-1}(x-\mu)\right].\]
Step 2: The factor \((2\pi)^{-p/2}|\Sigma|^{-1/2}\) does not involve \(x\); it is purely a normalizing constant that scales the total volume under the curve to 1. Changing it rescales the height of the surface uniformly but does not change the shape (the relative pattern of contours).
Step 3: The only part of the density that actually depends on \(x\), and therefore controls how the density rises and falls as \(x\) varies, is the quadratic form \((x-\mu)'\Sigma^{-1}(x-\mu)\) inside the exponential. Points with equal value of this quadratic form lie on the same density contour, and these contours are ellipsoids centered at \(\mu\), with axes and orientation fixed by \(\Sigma\).
Step 4: So it is the quadratic form \((x-\mu)'\Sigma^{-1}(x-\mu)\) that describes the shape (elliptical contours) of the multivariate normal density curve.
Final answer: option (C).