Stats - mle poisson distribution -- quick question

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
  • #1
binbagsss
1,254
11
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?

Thanks in advance.
 
Physics news on Phys.org
  • #2
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?

Thanks in advance.
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.

Similar threads

  • Calculus and Beyond Homework Help
Replies
21
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
Replies
2
Views
5K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
Back
Top