How to solve this Expectation of log of random variable

Question

This may seem a trivial Question but I am confused and never come across this kind of expression where I need to simplify for a function of a random variable $R$.

I have an expression $E\bigg [\frac{{(\log(R^p)})^2}{N} \bigg]$ where N = number of data points and is a constant; R = is a random variable which is vector of distances from a data point $x$ to its closest neighbor (k nearest neighbor distances); $p$ is the dimension of the data set = 2 in my case.

The way I have solved and got stuck later on is as follows:

$E\bigg [\frac{{(\log(R^p)})^2}{N} \bigg]$

$ = E\bigg [\frac{{(p\log(R)})^2}{N} \bigg]$

$ = E\bigg [\frac{{p^2(\log(R)})^2}{N} \bigg]$

$ = p^2E\bigg [\frac{{(\log(R)})^2}{N} \bigg]$

Then how do I proceed further? Please help

Answer 1

There are no general formulas for the expected value of a non-linear function of a random variable. Sometimes it may happen that the non-linear function has a known distribution itself, in which case its expected value has been derived by some good people in the past.

In your case, you are looking at

$$\frac {p^2}{N}E\left[(\log(R))^2 \right]$$

So you have to consider: What is the distribution of $R$? If it is known, then perhaps the distribution of its natural logarithm is also known, say $X = \log R$, or you can derive it. If it is known or you derive it, then maybe its mean and variance become available, $\mu_x, \sigma^2_x$. But then, the expected value you face is the second moment of $X$, and you can calculate it by the relation

$$\sigma^2_x = E(X^2) - \mu_x^2 \implies E(X^2) = \sigma^2_x + \mu_x^2$$

If this approach is intractable, you can explore the possibility of approximating the non-linear function by a linear expansion, although here you should consider the quality of the approximation.