# Stats - mle poisson distribution -- quick question

• binbagsss
Then the sum is now 1+1+1+3 = 6, and this is equal to the total sum of 9+5+7+8+9+9 = 47, and the ratio of 6/47 is the same as 1/1 * 5/47 + 1/1 * 7/47 + 1/1 * 8/47 + 3/1 * 9/47. So the two ways of calculating the sum are equivalent.In summary, the two methods of estimating the parameter λ for a Poisson distribution are equivalent: the maximum likelihood estimate is the same as the sum of the product of the number and its frequency divided by the total
binbagsss
This is probably a stupid question , but,

It's easy enough to show that the mle of a poission distribution is ## \bar{x}##: ## \hat{ \lambda}= \bar{x} ##

But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates ## \lambda ## for some data by ## \sum \frac{( number X frequency that number occurred)}{ n} ##

how are these 2 equivalent?

binbagsss said:
This is probably a stupid question , but,

It's easy enough to show that the mle of a poission distribution is ## \bar{x}##: ## \hat{ \lambda}= \bar{x} ##

But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates ## \lambda ## for some data by ## \sum \frac{( number X frequency that number occurred)}{ n} ##

how are these 2 equivalent?

It's the same thing. Suppose you have a sample 9, 5, 7, 8, 9, 9. You can get the sum by just adding them, or you can rewrite it as 1 x 5 + 1x 7 + 1 x 8 + 3 x 9.

## 1. What is a Poisson distribution?

A Poisson distribution is a statistical distribution that is used to model the probability of a certain number of events occurring within a fixed interval of time or space. It is often used to analyze data related to rare events or occurrences.

## 2. What is the maximum likelihood estimation (MLE) method?

The maximum likelihood estimation (MLE) method is a statistical method used to estimate the parameters of a probability distribution by finding the values that maximize the likelihood of the observed data. In the context of a Poisson distribution, MLE is used to estimate the value of the parameter lambda, which represents the average number of events occurring in a given interval.

## 3. How is a Poisson distribution different from a normal distribution?

A Poisson distribution is a discrete distribution, meaning that it is used to model data that can only take on whole number values. In contrast, a normal distribution is a continuous distribution, meaning that it can take on any value within a range. Additionally, a Poisson distribution is used to model rare events, whereas a normal distribution is used to model more common events.

## 4. What is the relationship between the mean and variance in a Poisson distribution?

In a Poisson distribution, the mean and variance are equal and are both equal to the parameter lambda. This means that the spread of data in a Poisson distribution is determined by the value of lambda, with larger values of lambda resulting in a wider spread of data and smaller values of lambda resulting in a narrower spread.

## 5. How is a Poisson distribution used in real-world applications?

Poisson distributions are commonly used in a variety of fields, including insurance, finance, and biology. They can be used to model events such as the number of accidents that occur in a given time period, the number of customers who arrive at a store in a given hour, or the number of mutations in a DNA sequence. They are also used in quality control to assess the number of defective products in a batch.

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