## Related questions with answers

The maternity ward at Dr. Jose Fabella Memorial Hospital in Manila in the Philippines 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. Sketch a graph of the probability distribution of X. c. What is the probability that the maternity ward will deliver three babies in one hour? d. What is the probability that the maternity ward will deliver at most three babies in one hour? e. What is the probability that the maternity ward will deliver more than five babies in one hour?

Solutions

VerifiedWe know that there are on average, 60 births in a day. We can find the average number of births in one hour as:

$\frac{60}{24}=2.5$

#### $\color{#c34632}{(a)}$

Variable X, defined as the number of births in an hour, has a Poisson distribution with parameter 2.5. The mean and standard deviation of a Poisson distribution with parameter $p$ are:

$\mu=p,\quad s=\sqrt{p}$

Thus, the mean of variable X is 2.5 and standard deviation is $\sqrt{2.5}=1.5811$

#### $\color{#c34632}{(b)}$

A graph of the probability distribution of X is:

As we know, the random variable $X$ represents the number of births in an hour. Based on the given information we can conclude that

$1)$ the value of the random variable $X$ can be any integer between zero and infinity.

$2)$ trials are independent, that is, the number of births in one hour does not depend on the number of births in another hour.

$3)$ the probability of birth in any time interval is the same for all intervals of equal size.

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