The energy consumption (with the proper units) of a city is normally distributed with $\mu=100$ and $\sigma=1$. After $\theta$ days another factory began to operate with unit consumption. What is the likelihood function for $x_1,\dots,x_n$ observations?
I think I should write down the joint probability of the consumption of the city and the consumption of the new factory, which I think has Poisson distribution with parameter $\theta$. But I'm not sure about that part. How should I interpret mathematically that the new factory began to operate after $\theta$ days?