Suppose that events are produced according to a Poisson process with an average of lambda events per minute. Each event has a probability $p$ of being Type A event, independent of other events.
Let the random variable $Y$ represent the number of Type A events that occur in a one-minute period. Prove that $Y$ has a Poisson distribution with mean $\lambda p$.
I read over it and I feel like I'm missing something because I still see it as having the mean as lambda not $\lambda p$.