Since the Yi's can only take on values of 0 or 1 with a set probability, they are Bernoulli trials. You can find the probability of having NO mutations on a plate by using the poisson distribution for k = 0 (k = # of occurances). Doing 1 - this probability gives you the probability of having at least 1 mutation on a plate. With this probability, you get your parameter for the Bernoulli RV's, and so you have the distribution of your Yi's.
Then, to find E(Z), just take the expectation inside of the summation (since expectations are linear operators). The Yi's are independent so you can find the expected value of each one and then sum them up from 1 to N to get the total expected value, E(Z).