6.8
Poisson Dağılımı
Poisson olasılık dağılımı ayrık bir olasılık dağılımıdır. Sabit bir zaman veya mekan aralığında, bilinen ortalama bir oranda ve son olaydan bu yana ge…
Poisson dağılımı, zaman, hacim, mesafe veya benzer herhangi bir birim gibi belirli bir aralıktaki olay oluşumlarına uygulanan bir tür ayrık olasılık dağılımıdır.
Günde ortalama üç olan günlük otomobil satışlarına ilişkin verileri düşünün. Bu durumda, Poisson dağılımı, ortalama değer etrafındaki yayılma derecesini tahmin etmek için kullanılabilir.
Örneğin, Poisson dağılımı formülünü kullanarak belirli bir günde tam olarak 4 araba satma olasılığını tahmin edebilir. Bu olasılık değeri, önceki satış geçmişine değil, yalnızca ortalamaya bağlıdır.
Aynı formülü kullanarak, daha iyi görsel temsil için diğer tüm olasılıklar hesaplanabilir ve çizilebilir.
Bir Poisson dağılımı için standart sapma, ortalamanın karekökü ile verilir.
Binom dağılımından farklı olarak, Poisson dağılımı örneklem büyüklüğü veya olasılıktan değil, yalnızca ortalamadan etkilenir. Ayrıca, Poisson dağılımındaki rastgele değişkenin üst sınırı yoktur.
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Q1: What types of events can the Poisson distribution model?
The Poisson distribution models discrete events occurring over a specified interval such as time, volume, or distance. It applies when events happen at a known average rate and independently of each other. Examples include daily car sales, misspelled words in a book, or any countable occurrences within a fixed interval where the average rate is known.
Q2: How does the Poisson distribution differ from the binomial probability distribution?
Unlike the binomial probability distribution, the Poisson distribution depends only on the mean and is unaffected by sample size or probability of individual events. The binomial distribution requires a fixed number of trials and constant probability, while Poisson applies to events in a continuous interval with no upper limit on the random variable.
Q3: What formula determines the standard deviation in a Poisson distribution?
The standard deviation of a Poisson distribution equals the square root of the mean. This relationship is unique to Poisson distributions and simplifies calculations since you need only the mean value to determine both the distribution's center and spread around that mean.
Q4: Can you use the Poisson distribution to approximate other probability distributions?
Yes, the Poisson distribution can approximate the binomial probability distribution when the probability of success is small (such as 0.01) and the number of trials is large (such as 1,000). This approximation simplifies calculations when binomial conditions are met but parameters make direct computation difficult.
Q5: Why does the Poisson distribution depend only on the mean?
The Poisson distribution models independent events occurring at a constant average rate within a fixed interval. Since each event is independent and the rate is known, only the mean is needed to describe the probability of any number of occurrences. Previous event history does not influence future probabilities in this model.
Q6: How would you calculate the probability of a specific number of events using Poisson?
The Poisson formula calculates the probability of exactly k events occurring given the mean. For example, with an average of three daily car sales, you can calculate the probability of selling exactly four cars on a given day. Once calculated, all other probabilities can be plotted in probability histograms for visual representation.
Q7: What real-world scenarios are suitable for Poisson distribution analysis?
Poisson distribution applies to countable events in fixed intervals where the average rate is known and events are independent. Examples include the number of misspelled words in 100 pages of a book, customer arrivals per hour at a service desk, or defects per unit of production. Any scenario with a predictable average rate over a defined interval fits this model.