Let the random variable Xn have a binomial distribution:

Where each Bi is independent and i distributed according to

We can look at X_n as the cumulated sum of a series of events that occur over time. The events are the individual Bi. Note that there are two parameters of interest here. Namely, the p and the n. The first governs the probability of each ?event? B, whereas the second governs the number of events.

The question is, what happens to the distribution of X_n as the number of events go to infinity? There arc two interesting cases, and the questions below relate to these.

(a) Suppose now, n ? ?, while p ? 0 such that ? = np remains constant. That is, the probability of getting a B_i = I goes to zero as is increases. But, the expected ?frequency? of getting a one remains the same. This clearly imposes a certain speed of convergence on the probability.

What is the probability Pr(X_n = k)? Write the implied formula as a function of p, n, and k.

(b) Substitute ? = np to write Pr(X_n = 4) as a function of the three terms shown in Question 1.

(c) Let n ? ? and obtain the Poisson distribution:

(d) Remember that during this limiting process, the p ? 0 at a certain speed. How do you interpret this limiting probability? Where do rare events fit in?

, - , Bi