Gamma Distribution

In this article, we will explore one of the fundamental probability distributions in statistics “ the Gamma distribution”. We’ll uncover its significance in modeling continuous random variables, contrasting it with the Poisson distribution. By understanding its parameters, such as shape and rate, and delving into practical examples like predicting patient arrivals at a dental office.

At its core, the Gamma distribution represents the probability of time until an event occurs, such as waiting times or the lifespan of certain products. It’s governed by two parameters: shape ( α) and rate ( β), which influence its shape and scale, respectively. (not to be confused with Poisson Distribution)

Poisson ?? vs Gamma Distribution ??

• Discrete vs continuous random variables: Poisson Distribution models the number of events occurring within a fixed interval of time, given the average rate at which events occur (ex: number of customers arriving). On the other hand, the purpose of the Gamma distribution is to model continuous random variables, representing non-negative real numbers, (such as waiting times, survival times…)
• Parameters: Poisson Distribution is defined by a single parameter, usually denoted as lambda λ (representing the average rate of occurrence of the events). Gamma Distribution is defined by two parameters: the shape ( α) and rate ( β)

Parameters of Gamma Distribution

1/ Shape parameter Alpha: The shape parameter describes how many events the distribution describes

2/ Scale parameter Beta: The scale parameters describe the time interval between the events we are modeling.

3/ Cumulative Distribution Function (CDF): The CDF of the Gamma distribution gives the probability that a random variable X is less than or equal to a certain value x. It is denoted as F(x) and is calculated as:

• γ(α,βx) is the lower incomplete gamma function.
• Γ(α) is the gamma function.

4/ The PDF:

The Probability Density Function (PDF) of the Gamma distribution is given by:

• x is the random variable.
• α and β are the shape and rate parameters.
• Γ(α) is the gamma function.

5/ The Expected Value

6/ The Variance

Efficient Dental Scheduling with Gamma Distribution

Consider a real-life scenario at a dental office. Patients arrive at the dental office according to a Poisson process (This implies that the arrivals of patients are independent of each other and occur randomly over time, without any fixed pattern or regular intervals) at an average rate of 15 per hour.

• Let’s calculate first the probability that it takes less than 10 minutes for the first 3 customers to arrive! ( we can use the gamma distribution since we are modeling waiting time in a Poisson process)

In that case :

• Shape parameter (α) (Number of arrivals) it’s 3.
• The scale parameter (β) is 60/15 or (1/15)*60

import scipy.stats as stats

alpha = 3
beta = 60/15 # Average time between arrivals (minutes)
time_until_3_customers = 10 

# Calculate the probability using the gamma CDF
probability_less_than_10_minutes = stats.gamma.cdf(time_until_3_customers, a=alpha, scale=beta)

print("Probability that it takes less than 10 minutes for the first 3 customers to arrive:", probability_less_than_10_minutes)        

The probability that it takes less than 10 minutes for the first 3 customers to arrive: 0.4561

• What is the average amount of time that will elapse before 3 customers arrive at the dental office?

mean = alpha*beta
mean #12        

The average amount of time until the arrival of the third customer is 12 minutes.

Note: If you don’t want to use Python, you can do the calculation here, with good data visualization too: https://homepage.divms.uiowa.edu/~mbognar/applets/gamma.html

As we wrap up our article, it’s clear that understanding the Gamma distribution is key for predicting events over time. We’ve seen how it differs from the Poisson distribution and how it’s used in real-life scenarios, like predicting patient arrivals at a dentist’s office. But here’s a thought to ponder! How can we use this knowledge to improve everyday processes even more? ??

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