Question

On an average weekend, I get three emails from students. I am interested in X, the random variable of the number of emails from students I will get this weekend.

1. What family of distributions does this situation belong to? What is (are) the parameter(s)?

2. What is the probability that I get one email, ?(?=1)?

3. What is the probability that I get 5 or less emails, ?(?≤5)?

4. I estimate that I will be able to answer all the emails as long as there are less than 25 of them. What is the probability that I will get more than 25 emails, ?(?>25)?

Answer 1

1) This situation belongs to the Poisson family of distributions       
The parameter λ of the Poisson distribution is λ = 3 emails per weekend       
        
2) Let X be the number of emails received on a weekend       
X follows a Poisson distribution with λ = 3 emails per weekend       
The pdf of the Poisson distribution is        
e-@* P(X = x) =

That is        

e-3(3)* P(X = x) = +1

To find P(X = 1)       
We use the Excel function POISSON.DIST to find the probability       
P(X = 1) = POISSON.DIST(1, 3, FALSE)               (Last parameter is FALSE for non-cumulative probability)   
                  = 0.1494       
P(you get one email) = 0.1494       
        
3) P(you get 5 or less emails)        
= P(X ≤ 5)       
We use the Excel function POISSON.DIST to find the probability       
P(X ≤ 5) = POISSON.DIST(5, 3, TRUE)                  (Last parameter is TRUE for cumulative probability)   
                 = 0.9161       
P(you get 5 or less emails) = 0.9161       
        
4) P(you get more than 25 emails)        
= P(X > 25)       
= 1 - P(X ≤ 25)       
We use the Excel function POISSON.DIST to find the probability       
= 1 - POISSON.DIST(25, 3, TRUE)       
= 1 - 1       
= 0       
P(you get more than 25 emails) = 0