Question

Suppose people get infected by Coronavirus according to a Poisson process with rate λ > 0 and

λ denoting the average number of infected people per day.

(a) (10 points) Find the expected time the 100th infected person will be identified.

(b) (10 points) Find the probability that the elapsed time between infected persons 10th and 11th exceeds two days.

Answer 1

We are given that people get infected with Coronavirus according to a Poisson process with rate λ > 0 and λ denoting the average number of infected people per day.

Let (i=1,2,...) denote the inter arrival times of the people infected with coronavirus (in days), i.e., X1 is the time (in days) at which a person infected with Coronavirus will be identified, X2 is the time (in days) between the identification of the second infected person and first infected person and so on. It means that Xi (i=1,2,...,n) is the time (in days) between the identification of the ith infected person and (i-1)th infected person.

Moreover, note that if we take:

then Sn (n=1,2,...) is the time (in days) at which the nth infected person will be identified.

Now, we know that if we have a Poisson process with rate λ per day, then the inter arrival times are independently and identically exponentially distributed with rate parameter λ per day. Thus, we get:

Thus, the PDF, CDF and Mean of Xi's is given by:

(a)

The time at which the 100th infected person will be identified is equal to S100, thus the expected time the 100th infected person will be identified is given by:

(b)

The elapsed time between infected persons 10th and 11th is equal to X11. Thus, the probability that the elapsed time between infected persons 10th and 11th exceeds two days is given by:

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