Poisson distribution (Exhibit 33.10) expresses the probability of a given number of events occurring in a fixed interval of time or space, if these events occur with a known constant rate and independently of the time since the last event.
The distribution is useful for modelling number of occurrences of an event over a period of time, for instances:
Properties:
A random variable X is said to be a Poisson random variable with parameter λ (> 0) if it has the probability function:
$$P(X=i)=\frac{e^{-λ} λ^i}{i!},\,for \,i \,(occurances)=0,1,2…$$ $$P(X=0)=\frac{e^{-λ} λ^0}{0!}= e^{-λ}$$ $$P(X=1)=\frac{e^{-λ} λ^1}{1!}= λe^{-λ}$$It can be shown that:
$$Mean \,E(X) = λ$$ $$Variance \,Var (X) = λ$$Thus, parameter λ can be interpreted as the average number of occurrences per unit time or space.
Example 1: Patients arrive at the A & E of a hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?
$$λ = \text{3 per 30 minutes}$$ $$P(X=4)=\frac{e^{-3} 3^4}{4!}=0.168 = 16.8 \% $$Example 2: Customers arrive at a row of ATMs at the average rate of 96 per hour during peak hours (weekday noon). What is the probability of 20 or more arrivals over 10 minutes on a weekday afternoon?
λ = 16 per 10 minutes. The Poisson distribution for λ = 16 is depicted in Exhibit 33.11.
$$P(X≥20)=1-P(X<=19)= 1-\sum_{i=0}^{19}\frac{e^{-16}16^i}{i!}=1-0.812=0.188$$There is 18.8% likelihood that 20 or more customers will arrive at the row of ATMs, over any 10-minute interval, on a weekday afternoon.
Note: Excel’s Poisson distribution function is POISSON.DIST (x, λ, cumulative).
Use the Search Bar to find content on MarketingMind.
In an analytics-driven business environment, this analytics-centred consumer marketing workshop is tailored to the needs of consumer analysts, marketing researchers, brand managers, category managers and seasoned marketing and retailing professionals.
Unlock the Power of Digital Marketing: Join us for an immersive online experience designed to empower you with the skills and knowledge needed to excel in the dynamic world of digital marketing. In just three days, you will transform into a proficient digital marketer, equipped to craft and implement successful online strategies.
Contact | Privacy Statement | Disclaimer: Opinions and views expressed on www.ashokcharan.com are the author’s personal views, and do not represent the official views of the National University of Singapore (NUS) or the NUS Business School | © Copyright 2013-2024 www.ashokcharan.com. All Rights Reserved.