Pertemuan 10 - Distribusi Poisson

Summary

This video discusses the concept of Poisson distribution, focusing on its application in calculating probabilities for discrete events occurring within a fixed time or space. The script explains how the Poisson distribution is used to model occurrences like phone call arrivals or errors in typed text, particularly when events are rare. It covers the formulas for calculating Poisson probabilities, introduces parameters like the mean (lambda), and provides examples of how to calculate the probability of various event occurrences. Additionally, the use of tables for Poisson distribution is explained, along with practical problem-solving techniques.

Takeaways

• 😀 The Poisson distribution is used to model the probability of a certain number of events occurring within a fixed time or area.
• 😀 Poisson distribution is helpful in scenarios like calculating the number of telephone calls, typing errors, or vehicles passing in a certain period.
• 😀 The Poisson distribution can also approximate the binomial distribution when the number of trials is large and the probability of success is small.
• 😀 The key parameter in a Poisson distribution is lambda (λ), which represents the average number of successes within a given time interval or area.
• 😀 The formula for calculating the Poisson probability is: P(X = x) = (e^(-λ) * λ^x) / x! where X represents the number of events and λ is the average rate.
• 😀 For the Poisson distribution, the possible values of the random variable X are 0, 1, 2, and so on.
• 😀 Variance and standard deviation for Poisson distribution are both equal to λ, meaning they are directly related to the average number of events.
• 😀 To calculate Poisson probabilities, either the Poisson formula can be used directly or a Poisson table can be consulted for the cumulative probability values.
• 😀 When calculating the probability of exactly X events, the cumulative probability from a table must be adjusted by subtracting the value for one less than X.
• 😀 In practice, problems often require finding the probability of events greater than a certain threshold. This can be calculated by subtracting the cumulative probability up to that point from 1.

Q & A

What is a Poisson experiment?

A Poisson experiment refers to a process that generates a random variable X, which represents the number of successes within a certain region or over a specific time interval. It is typically used to calculate the probability of events occurring within defined time units or lengths.

How is the Poisson distribution different from the binomial distribution?

The Poisson distribution can be used to approximate the binomial distribution when the number of trials (n) is large (greater than 30) and the probability of success (p) is small (less than 1). In contrast, the binomial distribution is used when there is a fixed number of trials with a constant probability of success.

What does the lambda (λ) or mu (μ) symbol represent in the Poisson distribution?

In the Poisson distribution, lambda (λ) or mu (μ) represents the average number of successes (or events) that occur within a specific time interval or region.

What is the general formula for the Poisson distribution?

The general formula for the Poisson distribution is P(X = x) = (e^(-λ) * λ^x) / x!, where e is the base of the natural logarithm, λ is the average number of successes, x is the number of occurrences, and x! is the factorial of x.

How can the Poisson distribution be used in real-life scenarios?

The Poisson distribution is useful for modeling events that occur at random within fixed intervals of time or space. Examples include the number of phone calls at a call center, the number of typing errors on a page, or the number of cars passing through a toll booth in a given time.

In the example of a telephone company, how do we calculate the probability that a specific choice is not selected?

The probability that a specific choice is not selected is calculated using the Poisson distribution formula. For example, if the probability of selection for each choice is 1/1000, we calculate the probability for x = 0, x = 1, x = 2, and x = 3, using λ = 0.2 for the number of selections.

How do you calculate the probability of x = 0, 1, 2, or 3 using the Poisson distribution?

To calculate the probability of x = 0, 1, 2, or 3, we apply the Poisson distribution formula for each value of x. For example, for x = 0, we use the formula P(X = 0) = e^(-0.2) * 0.2^0 / 0! and similarly for x = 1, 2, and 3.

What role does the Poisson distribution table play in solving probability problems?

The Poisson distribution table provides pre-calculated values of cumulative probabilities, which can be used to quickly determine the likelihood of a certain number of successes occurring. These values represent the probability that X is less than or equal to a certain number.

How do you use the Poisson table to calculate probabilities like P(X = 0, 1, 2)?

To calculate probabilities like P(X = 0, 1, 2), you need to look up cumulative probabilities for x ≤ 0, 1, and 2 in the Poisson table and subtract the cumulative probabilities of previous values. This gives the exact probability for the desired x value.

How can the Poisson distribution be used to predict the number of tanker trucks arriving at a port?

In the given example, where 10 tanker trucks arrive on average every day at a port, the Poisson distribution can be used to calculate the probability of more than 15 tankers arriving on any given day. This is done by calculating the cumulative probability for x ≤ 15 and subtracting it from 1 to find the probability of more than 15 tankers arriving.