Poisson Distribution

Introduction

TBA

Density function

The density function for the Poisson distribution is $$f(x | \lambda ) = \frac{e^{-\lambda}\lambda^x}{x!}$$

The density function for the Poisson distribution depends on the rate parameter $\lambda =$ $\, 1.5 \,$.

Specific to the plot above, the density function is $$f(x | \lambda = 1.5) = \frac { e^ { -1.5 } 1.5^x } { x! }$$

Examples

1. A seismologist has been studying the frequency of small earthquakes in a particular geologically active region. Over the course of several years, she determines that on average, there are 3.5 small earthquakes per day in this region.
   1. What is the probability that on a given day there are exactly 2 earthquakes?
   2. What is the probability that on a given day there are 3 or more earthquakes?
   3. Over the course of a week (7 days), what's the probability that there is at most 5 earthquakes?
2. You are a future math teacher preparing lesson plans on the Poisson distribution. You decide to use real-world data from the school library's study room booking system as a teaching tool. Records from the library indicate that, on average, 5 students request to book a study room every hour. Assuming that the number of booking requests follows a Poisson distribution:
   1. What is the probability that exactly 7 students will request to book a study room in a given hour?
   2. What is the probability that 3 or fewer students will request to book a study room in a given hour?
   3. During a two-hour span, what is the probability that more than 10 students will request to book a study room?
3. Consider a busy intersection in a city. Past data has shown that, on average, 5 vehicles per hour make a wrong turn at this intersection due to a lack of clear signage.
   1. What is the probability that exactly 3 vehicles will make a wrong turn in the next hour?
   2. What is the probability that no vehicles will make a wrong turn in the next hour?
   3. If the city's engineering department plans to monitor the intersection for a 4-hour period, what is the probability that more than 20 vehicles will make a wrong turn during this time?
4. In a chemical processing plant, a specific reactor is known to have a rare malfunction due to random impurities in the reactants. Over the past several years, data has shown that this malfunction occurs on average 1.5 times every 100 hours of operation. A new batch of reactant is being processed in this reactor for a continuous 48-hour run.
   1. What is the probability that there will be no malfunctions during this 48-hour run?
   2. What is the probability that there will be at least one malfunction during this 48-hour run?
   3. What is the probability that there will be more than two malfunctions during this 48-hour run?

Calculator

$X \sim$Poisson( $\lambda =$ $\, 1.5 \,$). $\mathbb{P}[X$ $\le$ $\, 5 \,$ $] = 1$.

References

Poisson distribution. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 06/10/2023.