Question

Laura and Andrea work as independent consultants in the area of human resources for companies that need training for their collaborators. The two give leadership conferences and do customer service training. It is estimated that the frequency with which they provide one or the other service can be framed within a Poisson process with different frequency rates: Laura gives leadership conferences with an average rate of 4 per month, while customer service trainings are performed at an average frequency of 9 every two months. On the other hand, it is estimated that in a month Andrea dictates, on average, 3 leadership conferences, while she is able to teach 72 customer service training courses per year, on average.
If necessary, assume a month is 4 weeks or 30 days. According to the previous information, answer:

a. What is the probability that in 18 weeks, Laura will teach 12 customer service courses and Andrea will give 10 leadership conferences?

Answer 1

Solution

Back-up Theory

If a random variable X ~ Poisson (λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!) ……..................................................................……..(1)

This probability can also be obtained by using Excel Function, Statistical, POISSON ….................................................... (1a)

If X = number of times an event occurs during period t, Y = number of times the same event

occurs during period kt, and X ~ Poisson(λ), then Y ~ Poisson (kλ) ………...............................................................…….. (2)

Now to work out the solution,

Let X1 = number of customer service trainings performed by Laura every two months. Then, given X1~ Poisson (9) ...... (3)

Let Y1 = number of leadership conferences given by Andrea in one month. Then, given Y1~ Poisson (3) ....................... (4)

Now, if

X = number of customer service trainings performed by Laura in 18 weeks or 4 ½ months .

Then, given vide (2), X~ Poisson (20.25) [9 in 2 months => 4.5 x 4.5 in 18 weeks] ............................................................ (5)

Y = number of leadership conferences given by Andrea in 18 weeks i.e., 4 ½ months.

Then, given Y~ Poisson (13.5) [3 in 1 month => 3 x 4.5 in 18 weeks]................................................................................. (6)

So, vide (1) and (5), the probability that in 18 weeks, Laura will teach 12 customer service

Courses, P(X = 12) = e ^{– 20.25}.20.25¹²/(12!)

= 0.0159 [vide (1a)].............................................................................................................................................................. (7)

Similarly, vide (1) and (6), the probability that in 18 weeks, Andrea will give 10 leadership conferences, P(Y = 10) = e ^{– 13.5}.13.5¹⁰/(10!)

= 0.0760 [vide (1a)].............................................................................................................................................................. (8)

Finally, given that Laura and Andrea work as independent consultants, vide (7) and (8), required probability

= 0.0159 x 0.0760

= 0.0012 Answer

DONE