Question

4. At a local hospital, babies are born at an average rate of four births per night (consider "night" to be an 8 hour period from 10pm to 6am). Assume that the time that elapsed from one birth to the next has the exponential distribution. Take note that we are concerned only with the rate at which babies are born, and we are ignoring the time spent in labor. We must also assume that the times spent between births are independent. This means that a particularly long delay between two births does not mean that there will be a shorter waiting period for the next one. We may then deduce that the total number of births during a time period has the Poisson distribution (a) Find the average time between two successive births. (b) Find the probability that after a birth occurs, the next birth occurs in less than an hour. (c)) Find the probability that exactly five births occur in a single night. (d) Find the probability that less than five births occur in a single night (e) Find the probability that more than 35 births occur in a week (7 nights),

Answer 1

We would be looking at the first 4 parts of the problem here as:

a) The average time between two consecutive births is computed here as:

= Total time for a night / Average number of births in a night

= 8/4 = 2

Therefore the average time required here is 2 hours.

b) Probability that after a birth occurs, next occurs in less than half an hour is computed here as:

= 1 - Probability of no birth in 30 minute period.

Average number of births in 30 minutes period = 4/16 = 0.25. Therefore the probability here is computed as:

Therefore 0.2212 is the required probability here.

c) Probability that exactly 5 births occur in a single night is computed here as:

$P(X = 5) = \frac{4^5}{5!}e^{-4} = 0.1563$

Therefore 0.1563 is the required probability here.

d) Probability that less than 5 births occur in a single night is computed here as:

$P(X < 5) = e^{-4} + 4e^{-4} + \frac{4^2}{2}e^{-4} + \frac{4^3}{3!}e^{-4}+ \frac{4^4}{4!}e^{-4} = 0.4405$

Therefore 0.4405 is the required probability here.