Expected Value and Variance – Calculating Expected Value and Variance of log(a)

Question

I have a random variable $X(a) = \log(a)$ where a is normal distributed $\mathcal N(\mu,\sigma^2)$. What can I say about $E(X)$ and $Var(X)$? An approximation would be helpful too.

Answer 1

If we consider "approximation" in a fairly general sense we can get somewhere.

We have to assume not that we have an actual normal distribution but something that's approximately normal except the density cannot be nonzero in a neighborhood of 0.

So let's say that $a$ is "approximately normal" (and concentrated near the mean*) in a sense that we can handwave away the concerns about $a$ coming near 0 (and its subsequent impact on the moments of $\log(a)$, because $a$ doesn't 'get down near 0'), but with the same low order moments as the specified normal distribution, then we could use Taylor series to approximate the moments of the transformed random variable.

For some transformation $g(X)$, this involves expanding $g(\mu_X + X-\mu_X)$ as a Taylor series (think $g(x+h)$ where $\mu_X$ is taking the role of '$x$' and $X-\mu_X$ takes the role of '$h$') and then taking expectations and then either computing the variance or the expectation of the square of the expansion (from which can be obtained the variance).

The resulting approximate expectation and variance are:

$\text{E}\left[g(X)\right]\approx g(\mu_X) +\frac{g''(\mu_X)}{2}\sigma_X^2$ and

$\text{Var}\left[g(X)\right]\approx \left(g'(\mu_X)\right)^2\sigma^2_X$

and so (if I didn't make any errors), when $g() = \log()$:

$$\text{E}\left[\log(a)\right]\approx log(\mu_a) -\frac{\sigma_a^2}{2\mu_a^2}$$

$$\text{Var}\left[\log(a)\right]\approx \sigma^2_a/\mu_a^2$$

* For this to be a good approximation you generally want the standard deviation of $a$ to be quite small compared to the mean (low coefficient of variation).