Problem 3: Binomial Distribution (30 points):
The population infection rate of COVID-19 in a given city at the end of May 2020 was 0.15.
In a random sample of 30 individuals from the population:
a)
What would be the expected (average) number of the infected patients?
a.
30*0.15 = 4.5
b)
What would be the standard deviation of the number of infected patients?
a.
30*0.15(1-0.15) = 3.825
c)
What is the probability that 10 patients out of 30 are infected?
a.
From StatCrunch, P(X = 10) = 0.0067
d)
If face masking reduces the population infection rate by 30%,
what is the probability that
10 out of 30 patients are infected?
a.
New infection rate = 0.15*0.7 = 0.105
b.
From StatCrunch, P(X = 10) = 0.00053
Problem 4: Poisson Distribution (30 points)
In the Poisson distribution, the average rate of a process (e.g., number of events per unit of time)
is approximately constant, i.e., as the number of events increases, the probability of occurrence
of any event is reduced, so the average rate remains constant, and the number of events that can
occur can vary from zero to a very large number (infinity).
The average rate of pandemics is estimated approximately 3 in 100 years
or 0.3 per 10 years.
a.
What is the probability that we will another pandemic in the next 10 years?
How about 2
pandemics in the next 10 years?
a.
From StatCrunch, P(X = 1) = 0.22
b.
From StatCrunch, P(X = 2) = 0.033
In a seminal paper by Richardson titled “The Distribution of Wars in Time”, Source: Journal of
the Royal Statistical Society, Vol. 107, No. 3/4 (1944), pp. 242-250
He tests the hypothesis that the number of conflicts per year is approximately constant: =
299/432 = 0.69, or approximately 7 conflicts per 10 years and it follows a Poisson distribution.
He defines conflicts as the one resulting in a number of deaths greater than a threshold (no need
to get into details here, those interested can read his article!).
b.
What is the probability of having zero conflicts in one year?
a.
From StatCrunch, P(X = 0) = 0.50
c.
What is the probability of having zero conflicts in 5 years?
a.
From StatCrunch, P(X = 0) = 0.030
