Question

Suppose that X ~ NB(r, p) [negative binomial distribution] and that Y ~ B(n, p) [binomial].

a. Give a probabilistic argument to show that P(X > n) = P(Y < r).

b. Use the FPF to express the equality in part (a) in terms of PMFs.

c. Using the complementation rule, how many terms of the PMF of X must be evaluated to determine P(X > n)?

d. How many terms of the PMF of Y must be evaluated to determine P(Y < r)?

e. Use your answers from part (c) and (d) to comment on the computational savings of using the result of part (a) to evaluate P(X > n) when n is large relative to r.

Answer 1

a) The event (X > n) means There are more than n trials required to get the rth success. Which means, if we consider the first n trials of the experiments then, the number of successes we obtained must be less than r., because otherwise, there wouldn't require more than n trials to get the rth success, which is equivalent to saying that (Y < r). Hence the probabilistic argument.

b) I don't know what is FPF though. If you can tell me the full form then, I will answer that. Otherwise the mathematical representation of the above argument will work.

c) There must be (n-r+1) terms to be calculated to determine P(X > n)

d) Only r terms need to be calculated to find P(Y < r)

e) When n is large relative to r, n-r >> r, so there are many more terms that needs to be calculated to find P(X > n), where as using (a), only r terms needs to be calculated, which is much less when n >> r