The restaurant owner also wants to reconfigure her seating layout, and has asked you for help in modeling her clients. She gives you a dataset of past reservations, and tells you that she gets a mix of single clients who come to sit at the bar, and people who come (alone or in groups) for dinner. You decide to model her clients using a mixture distribution, with fraction $\alpha$ of single bar patrons, and the remaining $(1-\alpha)$ fraction of diners; for the latter, you want to model the size of each group as $1+N_p$, where $N_p$ is Poisson distributed with parameter $\lambda$.
Compute the first and second moments.
The first moment is defined as $\mu_1 = E[X]$ and the second moment $\mu_2 = E[X^2]$. However, I am not sure what $X$ is in this case. Is it simply $X=1+N_p$, and then I take the expectation of this and the square of that same expression? In this case, I presume I have two equations for two unknowns and I solve for $\alpha$ and $\lambda$? Is this the case?