How do I find a confidence interval for a series of values that are log-normally distributed weighted by their values?
The distribution would look like:
$\sum_{k=1}^n 2^{mU_k}$
Where U is a random variable between 0 and 1, and n is the number of samples and m is some scale factor greater than 1.
I have found similar questions which have suggested working with mean and then multiplying by n (Confidence interval on sum of estimates vs. estimate of whole?), however this would not work with my distribution. There may often be large influxes of small numbers, which disappear when flattened into a histogram where each range is logarithmic (1 to 2, 2 to 4, 4 to 8, etc) and each bar is the sum, and so small numbers should not influence my distribution by more than their weight.
My weighted distribution looks like:
If all my samples were weighted equally (as opposed to by their value) the distribution would look like:
As you can see if I weight all points equally it makes the 8192 to 16384 range look equal in value to the 1024 to 2048 range, which in reality it is about 9 times more important. These small ranges cannot influence the distribution more than their sum.
Edit: U is a random variable between 0 and 1, and n is the number of samples.
