I am a little confused here on finding a probability but I can figure out the end I just need help on the intermediate steps.
So we have 55 independent normal observations, which all have the same mean $\mu$. The first 50 observations have variance $\sigma_1^2$ and the last 5 observations have variance $\sigma_2^2$.
What is the variance of the average of all 55 observations? I left out some of the numbers here so I can scrutinize it theoretically.
I've tried a few things.
- The sum of both variances. I ruled this out because one of the subsets has a much higher variance than the other, much smaller subset. I felt like this would somehow influence the ending variance/standard deviation, in the way that outliers do.
- The sum of both variances, weighted by the proportion of the entire set they make up. That is to say, if one of the subsets is 10/11ths of the data and the other is 1/11th, then their respective variances would take up that percentage of the variance of the average of all 55.
- The sum of both variances, divided by 55, the total number of all observations.
- The sum of both variances divided by $55^2$.
What about the average, while I'm asking? Do I add them together? Do divide $\mu$ by 50 and 5 respectively, and then add them? Do I add $\mu + \mu$ and then divide by 55? Is the average simply $\mu$?
Any help would be appreciated.