Question

1)                 The maternity ward at a certain hospital in a certain country in Africa is one of the busiest in the world with an average of 60 births per day. Let X = the number of births in an hour.

a.      Find the mean and standard deviation of X.

b.     What is the probability that the maternity ward will deliver three babies in one hour?

c.      What is the probability that the maternity ward will deliver at most three babies in one hour?

d.     What is the probability that the maternity ward will deliver more than five babies in one hour?

 

Answer 1

a.

Poisson distribution

$P(x=k)=\frac{\lambda^ke^{-\lambda}}{k!}$

mean:

$\lambda=E(X)=60/24=2.5$

standard deviation:

$\sigma=\sqrt{\lambda}=1.58$

b.

$P(x=3)=\frac{2.5^3e^{-2.5}}{3!}=0.2138$

c.

$P(x\le 3)=P(x=0)+P(x=1)+P(x=2)+P(x=3)=$

$=0.0821+0.2052+0.1283+0.2138=0.6294$

d.

$P(x>5)=$

$=1-(P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5))=$

$=1-(0.6294+0.1336+0.0668)=0.1702$