mean and variance of a biased sampled data

Question

Let said I have a normally distributed data of positive values.

Then, I sampled in this set and the probability of sample a value is proportional to the value. So bigger values will be sampled more frequently.

How can I estimate the mean and variance of the resampled data?

The data are inter-spike intervals, which are positive and distributed close to normal. When I randomly sample in time, longer intervals appear more frequently.

By simulations I got the following relationship: $\mu_{sample} = \mu *(1+ {\sigma}^{2}/\mu^{2})$. where $\mu_{sample}$ is the mean of the sampled data, and $\mu$, $\sigma$ are the mean and sigma for the dataset.

Answer 1

Let said I have a set of random positive values, like the steps length in a walk: $\{X_i\}_{i=1,N}$

The average step length is $\mu = \sum{X_i}/N$ and variance $\sigma^2 = \sum{(X_1}-\mu)^2 /N$

Now, the expected value of a random sample (in distance space, not in steps) is the sum of the probability of getting a step times is length: $\mu_{sample} = \sum P(X_i)*X_i$, where $P(X_i) = X_i/N\mu$

Whit some algebra it's possible to demonstrate that:

$\mu_{sample} = {1\over(N\mu)}\sum{X_i^2}$ = $\mu(1+{\sigma^2 \over \mu^2})$