Question

Suppose the number of admission to the emergency room at a small hospital follows a poisson distribution, but the incidence rate changes on different days of the week. On a weekday there are on average two admissions per day, while on a weekend day there is on average one admission per day.

Find the expected number of admissions for an entire week.

Answer 1

Let X1 is a random variable shows the number of admissions on 5 weekdays. Since number of admissions on weekdays is 2 per day and there are 5 weekdays so number of admissions on 5 weekdays is

Let X2 is a random variable shows the number of admissions on 2 weekends. Since number of admissions on weekends is 1 per day and there are 2 days on weekends so number of admissions on 2 days of weekends is

Now we have

Let T shows the total number of admissions for an entire week. So we have

T =X1+X2

Sum of Poisson random variables will be a Poisson random variable so

That is expected number of admissions for an entire week is 12.