I have a problem that goes like this: "The number of bicycle thefts in January this year is 214 which is 48 less then January the year before. Suppose the number of bicycle thefts are independent and poisson distributed and construct a 95% confidence intervall for the difference between the expectations of the two years".
So if I get things right i estimate the expectations with $\hat\lambda_1=262$ and $\hat\lambda_2=214$. Then $\hat\lambda_1 -\hat\lambda_2$ is normally distributed with mean $\hat\lambda_1 -\hat\lambda_2$ (?) but how should I estimate $\sigma^2$?
It's a long time since I did problems like this, so treat my answer with a pinch of salt.
I think what you are looking at is the distribution of the difference of two means.
If X and Y are the means of two independent samples, then the mean of their sum is the sum of the means, i.e. X+Y. Variance is the sum of the variances.
Similarly the mean of the difference is the difference of the means, i.e. X-Y. Somewhat confusingly, but logically, as we are dealing with squares, the variance is the sum of the variances.
That should let you construct your confidence interval.