I'm dealing with a food stand salesman. He notices that customers stop at his food stand with an average rate of $10$ per hour and he runs his food stand for about $8$ hours a day. Customer's arrive independently and are equally likely to arrive any time the store is open. They (independently of each other and of the arrival times of the customers) select either hotdogs or pretzels. $\frac{1}{4}$ of them choose the pretzel, and $\frac{3}{4}$ of them choose the hotdog. I'm looking to find the expected number (and variance) of hotdogs this salesman will sell on a typical day.
I'm trying to find a way to model this, but I'm drawing a lot of blanks. I've got a feeling this seems like a Poisson distribution, but I'm not sure how the different alternatives will factor into this.
A Poisson model is reasonable. The number of customers in a day has Poisson distribution with parameter $80$. The number of hot dogs sold in a day therefore has Poisson distribution with parameter $\lambda=\frac{3}{4}\cdot 80$. Now recall that a Poisson random variable with parameter $\lambda$ has mean and variance $\lambda$.