Poisson Distribution | Theory with examples

Summary

This video explains the Poisson distribution in detail, highlighting its theoretical basis and various properties. The distribution is introduced with examples, such as calculating the probability of defective items in a factory or errors in printed pages. Key concepts include the relationship with binomial distribution, the nature of small probabilities, and the formula used for solving related problems. It also covers important characteristics like the mean, variance, and standard deviation of the distribution, alongside an exploration of the distribution's shape and its dependency on the parameter value. The Poisson distribution’s practical application is illustrated through real-life examples.

Takeaways

• 😀 Poisson distribution is a theoretical probability distribution used when the probability of an event happening is very small.
• 😀 Similar to the binomial distribution, Poisson distribution is part of the discrete probability distribution family.
• 😀 The Poisson distribution is typically applied in situations where events occur at a very small probability, such as defective items produced by a factory or typing errors in printed pages.
• 😀 The distribution is named after the French mathematician, Simon-Denis Poisson, who developed it in 1837.
• 😀 The Poisson distribution is defined by certain properties, including its discrete nature and its basis in the Poisson theorem of logarithms.
• 😀 The distribution's key parameter, 'm', represents the mean of the distribution, which is essential for solving questions involving Poisson distribution.
• 😀 In a Poisson distribution, the number of trials can extend infinitely, meaning 'n' can approach infinity.
• 😀 Poisson distribution is useful in calculating the probability of very rare events, where the probability approaches zero, such as defects in a machine or earthquakes in a region.
• 😀 The probability of success and failure in Poisson distribution approaches 0 and 1 respectively, where the likelihood of an event happening becomes very small.
• 😀 The distribution’s formula involves the use of an exponential function 'e' (approximately 2.718), with the key formula being: (e^(-m) * m^x) / x! where 'x' is the number of successes.
• 😀 One important characteristic of Poisson distribution is its symmetry, meaning the mean and variance are equal, and its distribution shape is influenced by the value of 'm'.

Q & A

What is Poisson Distribution?

Poisson Distribution is a theoretical probability distribution used to model situations where the probability of an event happening is very small. It is used when the event occurrences are rare within a fixed interval or space.

What are the key differences between Poisson and Binomial distributions?

The Poisson distribution is used for events with a very small probability of occurrence, while the Binomial distribution is used for events with a fixed probability over a fixed number of trials. Poisson is continuous, whereas Binomial is discrete.

Where is Poisson Distribution typically used?

Poisson Distribution is used in situations where the probability of an event occurring is very small, such as the probability of producing defective items in a factory, typing errors on a printed page, or earthquakes occurring in a country in a year.

Who developed Poisson Distribution, and when?

Poisson Distribution was developed by French mathematician Simeon-Denis Poisson in 1837, and the distribution is named after him.

What are the main properties of Poisson Distribution?

Poisson Distribution is a discrete probability distribution based on the Poisson theorem of logarithms. It is used in situations where the number of trials can extend to infinity, and where the probability of success is very small.

What is the formula used in Poisson Distribution?

The formula for Poisson Distribution is: P(X = x) = (e^-λ * λ^x) / x!, where λ is the mean number of occurrences, and x is the number of occurrences we are interested in.

What is the significance of 'λ' (lambda) in Poisson Distribution?

In Poisson Distribution, 'λ' (lambda) represents the mean number of occurrences within a specified interval. It is a key parameter that helps define the distribution.

How do you calculate the probability of exactly 'x' successes in Poisson Distribution?

To calculate the probability of exactly 'x' successes in Poisson Distribution, you apply the Poisson formula: P(X = x) = (e^-λ * λ^x) / x!, where 'λ' is the mean and 'x' is the number of successes.

What is the relationship between the mean and variance in Poisson Distribution?

In Poisson Distribution, the mean and variance are equal. The value of both the mean (λ) and the variance is calculated as 'λ'. This is an important characteristic of Poisson distribution.

How is the shape of the Poisson Distribution graph determined?

The shape of the Poisson Distribution graph depends on the value of 'λ'. If 'λ' is small, the distribution is highly skewed, and if 'λ' increases, the distribution becomes more symmetric, resembling a normal distribution.