Let the random variable Xn have a binomial distribution:
align="center">

where each Bi is independent and is distributed according to

We can look at Xn as the cumulated sum of a series of events that occur over time. The events are the individual Bi. There are two parameters of interest here. Namely, the p and the n. The first governs the probability of each "event" Bi, whereas the second governs the number of events. The question is, what happens to the distribution of Xn as the number of events go to infinity? There are two interesting cases, and the questions below relate to these.
(a) Suppose now, n†’ˆž, while p †’ 0 such that λ = np remains constant. That is, the probability of getting a Bi = 1 goes to zero as n 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 P(Xn = k)? Write the implied formula as a function of p, n, and k.
(b) Substitute λ = np to write P(Xn = k) as a function of the three terms shown in Question 1.
(c) Let n†’ˆžand obtain the Poisson distribution:

align="center">

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?

Transcribed Image Text:

1 with probability p 0 with probability 1 -p