Distribusi Poisson | Materi dan Contoh Soal

Summary

In this lecture, the Poisson distribution is introduced as a discrete probability distribution used to model random events occurring within a fixed time period or specific area. The video explains its key properties, such as independence of events, the relationship with time or area, and the formula for calculating probabilities. Examples include calculating the likelihood of events like school closures or phone calls within a time frame. The lecture also demonstrates how to apply Poisson’s cumulative distribution and formula to real-world problems, making complex statistical concepts accessible and applicable to various scenarios.

Takeaways

• 😀 Poisson experiments involve counting the number of events occurring in a fixed interval of time or space.
• 😀 The results of a Poisson experiment are discrete and depend on the time or area in which they are measured.
• 😀 Key examples of Poisson experiments include phone calls per hour, school closures due to a virus, and rats in a field.
• 😀 The Poisson distribution formula for calculating probabilities is P(X = x) = (e^(-μ) * μ^x) / x! where μ is the mean number of occurrences.
• 😀 In Poisson distribution, events occur independently, meaning the occurrence of one event does not affect others.
• 😀 Poisson distribution can be used to calculate probabilities for specific outcomes, like the likelihood of a certain number of school closures or rats in a field.
• 😀 Cumulative Poisson probability helps determine the likelihood of multiple events happening within a range, such as P(X ≤ 6).
• 😀 A key property of Poisson distribution is that the probability of events occurring in very short intervals or small areas is proportional to the length or size of the interval or area.
• 😀 The average number of occurrences (μ) is crucial in determining probabilities in Poisson distribution.
• 😀 The transcript explains how to use the Poisson formula and cumulative tables for solving problems, such as calculating the probability of more than a certain number of events.
• 😀 Example calculations using Poisson distribution show how to determine probabilities for various scenarios like school closures or rats in a field, with step-by-step formulas provided.

Q & A

What is the definition of a Poisson experiment?

A Poisson experiment is a type of probability experiment that produces a discrete random variable, X, which represents the number of occurrences of an event during a specific time interval or within a defined area. The occurrences are independent of each other.

What types of intervals or areas are used in a Poisson experiment?

In a Poisson experiment, the time intervals can range from minutes to days, months, or even years. The defined areas could be a specific land area, volume, or length of a line where the event is measured.

What are the key properties of Poisson experiments?

The three key properties of Poisson experiments are: 1) The number of events in disjoint intervals or areas is independent; 2) The probability of an event in a small interval or area is proportional to the length of the interval or the size of the area; 3) The probability of multiple events occurring in a very small interval or area is negligible.

What does the Poisson distribution describe?

The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time or a fixed area, based on the average rate of occurrence of the event.

What is the formula for calculating the probability in a Poisson distribution?

The probability of observing X events in a Poisson distribution is given by the formula: P(X = x; μ) = (e^(-μ) * μ^x) / x!, where μ is the average number of events, and x is the number of events observed.

What is the significance of the constant 'e' in the Poisson distribution formula?

The constant 'e' (approximately 2.71828) represents the base of the natural logarithm and appears in the Poisson distribution formula as part of the exponential function, capturing the rate at which events occur in a given time interval or area.

How do you calculate cumulative probabilities in a Poisson distribution?

To calculate cumulative probabilities in a Poisson distribution, you sum the individual probabilities for all values of X from 0 to the desired number of occurrences. This can also be done using Poisson cumulative distribution tables.

What is an example of a Poisson experiment involving phone calls?

An example of a Poisson experiment involving phone calls is measuring the number of phone calls received by a company in an hour. This involves a time interval (an hour) and a discrete number of events (calls).

How do you use a Poisson cumulative distribution table?

To use a Poisson cumulative distribution table, find the value of λ (the average rate of occurrences), then locate the corresponding probability for the desired number of events. For cumulative probabilities, subtract the probability for the number of events that are not included.

What is the probability of a school being closed for exactly 6 days given an average closure rate of 4 days?

Using the Poisson distribution formula with μ = 4 and x = 6, the probability is calculated as P(X = 6) ≈ 0.1042, meaning there is approximately a 10.42% chance of the school being closed for exactly 6 days.