A call center wants to analyse customer behavior and counts on $n$ days incoming calls. Which statistical model is adequate in this situation? Define an unbiased estimator regarding incoming calls.
We make the following reasonable assumptions:
1.) The probability that an incoming call arrives during an arbitrary small time period (during office hours) is the same for each time period.
2.) Further, the calls that arrive on one day are independent
3.) The number of calls that arrive on day are independent from the numbers of the other $n$ days.
So it is reasonable to assume a Poisson distribution $\pi_{\lambda}$ for each day, where $\lambda\geq 0$ denotes the Poisson paramter (we also know that is the expectation). This delivers the statistical model $$ (\mathbb{N}^n,\mathcal{F}(\mathbb{N}^n), \prod\limits_{i=1}^n\pi_{\lambda}), $$ where $\lambda$ is the unknown parameter which needs to be estimated. We can estimate $\lambda$ simply by taking the first observation of the $n$-many observations. We immediately see that $\mathbb{E}(x_1)=\lambda$ so it is unbiased.
Is this correct?