# Gamma Distribution

**Episode #10 of the course Theory of probability by Polina Durneva**

Good morning!

Today is our final lesson, and we will complete the course by talking about gamma distribution! Gamma distribution is a generalized form of exponential distribution and is used to model waiting times. Gamma distribution is also highly useful if you want to model time before event r happens.

**What Is Gamma Distribution?**

Once again, gamma distribution is used in modeling waiting times. For example, if you want to model lifespan until death, you will use gamma distribution.

The main assumptions for gamma distribution is the same as those for exponential and Poisson distributions:

1. The intervals over which the events occur do not overlap.

2. The events are independent.

3. The probability that more than one event happens in a very short time period is approximately 0.

**Formula for Gamma Distribution**

The formula for gamma distribution is probably the most complex out of all distributions you have seen in this course. The formula takes the following form: P(x) = (λ^{k}x^{k-1}e^{-λx}) / Г(x), where λ is the rate, or the frequency, of events, k is called the shape parameter, and Г(x) is the gamma function and is equal to (x – 1)!

The applications of gamma distribution mainly take place in queuing models for supply chain designs or modeling climatic conditions. Because of high complexity of applications of this type of distribution, we will not cover any examples here, as it would take a lot of time.

**Expected Value and Variance of Gamma Distribution**

The expected value of gamma distribution can be calculated by multiplying λ by k (the rate by the shape parameter). Moreover, the variance can be derived by multiplying λ^{2} by the shape parameter k.

**Other Continuous Probability Distributions**

Even though this course is nearly over, I encourage you to take a look at the following continuous probability distributions:

• **Beta distribution** is used to model probabilities of successes.

• **Normal distribution** is used to represent random variables whose distributions are not known.

• **Logistic distribution** is used in machine learning tools such as neural networks.

• **Pareto distribution** is mainly used in econometrics to model incomes of population.

• **Dagum distribution** is also used to model income and wealth in population.

**What’s Next?**

We have officially finished our journey in the basics of the theory of probability. We have started the course by talking about the basic addition and multiplication rules in probability, dependent and independent events, mutually exclusive events, and conditional probability. Then, we proceeded to more complicated topics about discrete probability distribution and continuous probability distributions.

I hope your journey does not stop here and you will continue exploring this wonderful and magnificent world of random events and ways to model them as accurately as possible. There is much more to discover and learn in this truly amazing world of probabilities and their applications in real world!

Thank you for staying with me over the past ten days, and I can’t wait to see you again!

Thank you again,

Polina

**Recommended book**

*Mathematical Fallacies and Paradoxes* by Bryan Bunch

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