AdvancedTopic 81 of 104Engineering Mathematics
Binomial and Poisson distributions
Model counts of successes in trials and rare events such as failures or arrivals.
In depth
The binomial distribution counts successes in n independent trials with probability p: P(k)=C(n,k)pᵏ(1-p)ⁿ⁻ᵏ, mean np. The Poisson distribution with rate λ models rare events: P(k)=λᵏe^{-λ}/k!, mean and variance λ.
Key points
- Poisson approximates binomial for large n, small p
- Defects per metre, calls per minute
- Events independent of each other
ExampleIf a server gets 3 failures per year, P(no failure) = e⁻³≈0.05.
Counting models describe failures, arrivals and successes, so they are used for quality control, queueing and reliability.
What is Binomial and Poisson distributions?
Model counts of successes in trials and rare events such as failures or arrivals.
Why is it important to understand binomial and Poisson distributions?
Counting models describe failures, arrivals and successes, so they are used for quality control, queueing and reliability.
When should I use a Poisson distribution?
Use it for counts of rare, independent events in a fixed interval, where the average rate is known and constant.
What is the difference between binomial and Poisson?
A binomial counts successes in a fixed number of trials. A Poisson counts events over time or space without a fixed number of trials.
How is Binomial and Poisson distributions used in practice?
Typical applications include estimating the number of defective items in a sample of 50, predicting the number of call arrivals per minute at a help desk and modelling the number of failures of a machine in a month.
What should I learn before Binomial and Poisson distributions?
Study "Expectation and variance" first (topic 80 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Binomial and Poisson distributions?
Poisson approximates binomial for large n, small p. Defects per metre, calls per minute. Events independent of each other.
Can you give an example of Binomial and Poisson distributions?
If a server gets 3 failures per year, P(no failure) = e⁻³≈0.05.
What comes after Binomial and Poisson distributions?
The next topic is "Normal distribution", which builds directly on this one.
How difficult is Binomial and Poisson distributions?
It is classed as Advanced in Engineering Mathematics (topic 81 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.