Question: Poissonization of the binomial distribution, Let N have Poisson (λ) distribution, Let X be a random variable with the following property: for every n, the conditional distribution of X given (N = n) is binomial (n, p).
a) Show that the unconditional distribution of X is Poisson, and find its parameter, it I s known that X-rays produce chromosome breakages in cells. The number of such breakages usually follows a Poisson distribution quite closely, where the parameter depends on the time of exposure, et. For a particular dosage and time of exposure, the number of breakages follows the Poisson (0.4) distribution. Assume that each breakage heals with probability 0.2, independently of the others.
b) Find the chance that after such an X-ray, there are 4 healed breakages.