Say we have a single observation x from a rv X with distribution Pois(\(\lambda\)+b), where b is known. Apply each of the following methods to find 95% intervals when x=23 and b=5.6.

  1. Find a \((1-\alpha)100\%\) confidence interval for \(\lambda\) by inverting the large-sample LRT.

  2. Find a \((1-\alpha)100\%\) confidence interval for \(\lambda\) by inverting the LRT without using the large sample theory.

  3. Find a \((1-\alpha)100\%\) credible interval for \(\lambda\) by using the prior \(\pi(\lambda)=1\).