The Multivariate Gaussian pdf is given by

$$(2\pi)^{-\frac{K}{2}} \det(\Sigma)^{-\frac{1}{2}} \exp({-\frac{1}{2}}(X-\mu)' \Sigma^{-1} (X-\mu)) $$

The wikipedia for multivariate Gaussians is here

However I could not find a pdf for the multivariate lognormal distribution. Does it exist? If so, what is it?


1 Answer 1


Just for the sake of completeness, I'll provide an answer here. This is a simple application of the multivariate change of variables theorem: say $Y = \Phi(X)$ where $\Phi$ is a smooth bijective function. Then

$$ p_Y(y) = p_X(\Phi^{-1}(y)) \vert \det J_{\Phi^{-1}}(y)\vert $$ where $J_{\Phi^{-1}}$ is the Jacobian matrix of the inverse transformation. So for a multivariate lognormal random variable $Y = \exp(Z)$ where $Z\sim \mathcal{N}(\mu,\Sigma)$, we have $\Phi^{-1}(y) = \ln(y)$ and so

$$ J_{\Phi^{-1}}(y) = \text{diag}(1/y_1,\ldots,1/y_n) $$ Hence

$$ \vert \det J_{\Phi^{-1}}(y)\vert = \prod_{j=1}^ny_j^{-1} = \frac{1}{y_1y_2\cdots y_n} $$ So, finally we have

$$ p_Y(y) = (2\pi)^{-n/2}(\det\Sigma)^{-1/2}\prod_{j=1}^ny_j^{-1}\exp\left(-\frac{1}{2}(\ln(y)-\mu)^t\Sigma^{-1}(\ln(y) - \mu)\right) $$

  • $\begingroup$ just in case anyone is interested, the entropy of this multivariate log-normal is $$H(Y) = \frac{n}{2}(1+ \log(2\pi)) + \log(\det(\Sigma))/2 + \sum_j \mu_j$$ $\endgroup$
    – sefi
    Jun 7, 2023 at 8:43

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