Hi guys I have been going through a bank of revision questions in prep for an exam and have come across this question that I cannot seem to solve. My work so far:
For part a, am I correct in assuming that since both distributions are independent that you take the log likelihood functions of both distributions and then add them together? With the log likelihood being $-\frac{n}{2} \ln(2π)-\frac{n}{2}\ln(σ^2)-\frac{1}{2σ^2}\sum_{j=1}^n(x_j -μ)$ for $x$.
For part b I am more unsure about as to what to do, any help would be appreciated. Thanks
You are right that you add those, but since it was assumed to be known that $\mu=0,$ you can just drop $\mu.$ Also, you didn't square the term in the sum. Where you had $\sigma$ you need $\sigma_1,$ and then when you do this with $m$ rather than $n$ you need $\sigma_2.$ For the likelihood under the null hypothesis you won't need to estimate two different parameters.