if $x = (x_1,x_2,...,x_n)$ and $x \sim N(\mu,\sigma\cdot I_n)$. How to calculate or estimate the mean and variance of $L_p(x)/L_q(x)$, where the $L_p(x)$ means: $$L_p(x) = (\sum_{i=1}^n|x_i|^p)^{\frac{1}{p}}.$$
for example, when $p=2$ and $q=1$, i want to calculate or estimate $E(\frac{L_2(x)}{L_1(x)})$ and $VAR(\frac{L_2(x)}{L_1(x)})$. BTW, for my numerical experiments, I found that the $VAR(\frac{L_2(x)}{L_1(x)})$ is very close to zero. and the value of $E(\frac{L_2(x)}{L_1(x)})$ may depend on $n$.