Poisson Distribution EXPLAINED in UNDER 15 MINUTES!

Summary

This video script provides an in-depth explanation of the Poisson distribution, a discrete probability distribution used to model the number of events occurring within a fixed interval. Named after French mathematician Siméon Denis Poisson, the distribution requires only one parameter, Lambda, representing the expected number of events per interval. The script discusses the assumptions behind the distribution, such as constant event rate and event independence, and demonstrates how to calculate probabilities using Excel's Poisson functions. It also explores the distribution's properties, including its mean and variance both being equal to Lambda, and concludes with a practical example involving Facebook ad click-throughs, questioning the real-world applicability of the Poisson distribution in scenarios with variable event rates.

Takeaways

• 📊 The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed time interval or region of opportunity, such as the number of customers a bank teller serves per hour.
• 🔢 It requires only one parameter, Lambda (λ), which represents the expected number of events per time interval and is also the distribution's mean and variance.
• 📉 The distribution is theoretically unbounded, extending from zero to infinity, though in practice, probabilities for high event counts become negligible and are often not plotted.
• ⏲ The Poisson distribution assumes a constant rate of event occurrence, meaning the probability of an event in any given time interval is the same as for any other interval of the same length.
• 🤝 The events are assumed to be independent, so the occurrence of one event does not affect the likelihood of subsequent events.
• 📚 The probability mass function (PMF) of the Poisson distribution can be calculated using a formula involving Lambda and the number of events (x), or with the help of statistical functions in software like Excel.
• 📈 Excel's POISSON.DIST function can be used to calculate the PMF for a given number of events, with arguments specifying the event count, the mean (Lambda), and whether to return the PMF or the cumulative distribution function (CDF).
• 📊 The CDF of the Poisson distribution can be used to find the probability of an event count being less than or equal to a certain number, and can be calculated using the POISSON.DIST function with the CDF argument set to true.
• 🌐 The distribution's shape varies with Lambda, with higher Lambda values leading to a distribution that more closely resembles a normal distribution due to the Central Limit Theorem.
• 💡 The appropriateness of using the Poisson distribution depends on whether the assumptions of constant event rate and event independence hold true for the scenario in question.
• 📝 In the given example, the number of click-through sales from a Facebook ad is assumed to follow a Poisson distribution with a mean of 12 sales per day, but the assumptions may not hold due to varying user activity on Facebook throughout the day.

Q & A

What is the Poisson distribution named after and what does it describe?

The Poisson distribution is named after a French mathematician, Siméon Denis Poisson. It describes the number of events occurring in a fixed time interval or region of opportunity, such as the number of customers a bank teller gets every hour.

Is the Poisson distribution discrete or continuous?

The Poisson distribution is discrete, meaning it can only take a specific set of values.

What is the only parameter required for the Poisson distribution?

The Poisson distribution requires only one parameter, Lambda (λ), which is the expected number of events per time interval.

How does the Poisson distribution differ from the binomial distribution in terms of event occurrence?

The Poisson distribution assumes that the events are independent and the occurrence of one event does not affect the occurrence of subsequent events, unlike the binomial distribution which has a fixed number of trials.

What are the assumptions underlying the Poisson distribution?

The assumptions underlying the Poisson distribution are that the rate at which events occur is constant and that the events are independent, meaning the occurrence of one event does not influence the next.

How can you calculate the probability mass function (PMF) for the Poisson distribution?

The PMF for the Poisson distribution can be calculated using the formula with the given value for x (number of events) and Lambda (λ), the mean or expected number of events per time period.

What is the relationship between the expected value and variance in the Poisson distribution?

In the Poisson distribution, the expected value (mean) is equal to its variance. Both are represented by Lambda (λ).

How can you use Excel to calculate the PMF of the Poisson distribution?

In Excel, you can use the POISSON.DIST function with three arguments: the number of events for which you're seeking the probability, the mean (Lambda), and a boolean indicating whether you want the cumulative distribution function (CDF) or the PMF (use FALSE for PMF).

What is the cumulative distribution function (CDF) in the context of the Poisson distribution?

The CDF in the Poisson distribution is the sum of the probabilities of all events up to and including a certain number. It can be calculated using the POISSON.DIST function in Excel with a TRUE argument for the third parameter.

How might the Poisson distribution be used to model click-through sales from a Facebook ad?

The Poisson distribution can be used to model the number of click-through sales from a Facebook ad by treating each click-through as an event and using the average number of click-through sales per day as the Lambda (λ) value.

Why might the Poisson distribution not be entirely appropriate for modeling click-through sales from a Facebook ad?

The Poisson distribution assumes a constant rate of event occurrence, which may not hold true for Facebook ad clicks due to varying user activity levels at different times of the day.

What is the probability of getting exactly 10 click-through sales in the first day, given a mean of 12?

The probability of exactly 10 click-through sales can be calculated using the Poisson PMF with Lambda (λ) equal to 12 and x equal to 10, resulting in approximately 10.5%.

How can you calculate the probability of getting at least 10 click-through sales in the first day?

To find the probability of at least 10 click-through sales, use the CDF to find the probability of getting 9 or fewer and subtract it from 1, which gives the probability of 10 or more.

What is the probability of more than one click-through sale in the first hour, given a mean of 12 sales per day?

Since the mean number of sales per hour would be 0.5, you would use a Poisson distribution with Lambda (λ) equal to 0.5 and calculate the probability of more than one sale, which is approximately 9%.