Confidence interval for a function of the MLE

Question

I am studying an old assignment in which I have calculated the MLE for a sample from an exponential distribution. It then gives the formula for the median of an exponential distribution $\ln(2)/\lambda$ and asks for the 95% confidence interval for the median.

I think for the MLE the expected information can be used to get the variance, but what about the function $\ln(2)/\lambda$?

Should I look at the Delta method or can the lower bound for the MLE confidence interval by substituted into the function as in $\ln(2)/\text{LCL(MLE)}$, $\ln(2)/\text{UCL(MLE)}$?

Answer 1

gives the formula for the median of an exponential distribution $\ln(2)/λ$

... so we're dealing with the rate-parameterization of the exponential.

and asks for the 95% confidence interval for the median. I think for the MLE the expected information can be used to get the variance, but what about the function $\ln(2)/λ$?

Should I look at the Delta method or can the lower bound for the MLE confidence interval by substituted into the function as in ln(2)/LCL(MLE), ln(2)/UCL(MLE)?

You shouldn't need the delta method to produce an interval for $\ln(2)/\lambda$.

Think about the probability statement a confidence interval for $\lambda$ makes.

Convert it to a probbaility statement about $\ln(2)/\lambda$.