Relationship b/w Binomial, CLT & Poisson Distrib.

In summary, the central limit theorem states that the binomial distribution can be approximated by a normal distribution N(0,1). However, it can also be approximated by a Poisson distribution. The relationship between the normal and Poisson distributions depends on how the limit is taken, with the normal distribution using a constant p and the Poisson distribution using a constant np as n approaches infinity. The standardisation in the central limit theorem also plays a role in the difference between the two distributions.
  • #1
azay
19
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From the central limit theorem the binomial distribution can be approximated by a normal distribution N(0,1). But the binomial distribution can also be approximated by a poisson distribition.

Does this mean there is a relationship between the normal distribution and the poisson distribution (especially as n->infinity in B(n,p))?

I'm confused about this. To me it sounds like the normal distribution and the poisson distribution are equal then. Or does the standardisation in the central limit theorem make the difference?

Thanks
 
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  • #2


The difference depends on how you take the limit. To get a normal from binomial, you keep p constant as n -> ∞. To get a Poisson from a binomial, you keep np constant as n -> ∞.
 

1. What is the relationship between the Binomial, Central Limit Theorem (CLT), and Poisson Distribution?

The Binomial, CLT, and Poisson Distribution are all related to each other in the sense that they are all probability distributions used to model discrete data. The Binomial Distribution is used to model the number of successes in a fixed number of trials, the CLT is used to approximate the distribution of sample means from any population, and the Poisson Distribution is used to model the number of events occurring in a fixed time or space. They are all interrelated through their underlying assumptions and properties.

2. How does the Central Limit Theorem relate to the Binomial Distribution?

The Central Limit Theorem states that as the sample size increases, the distribution of sample means will approach a normal distribution regardless of the shape of the underlying population distribution. This means that for a large enough sample size, the Binomial Distribution can be approximated by a normal distribution, making it easier to calculate probabilities and make inferences about the data.

3. Can the Poisson Distribution be used to approximate the Binomial Distribution?

Yes, the Poisson Distribution can be used to approximate the Binomial Distribution when the number of trials is large and the probability of success is small. This is because the Poisson Distribution assumes a fixed rate of occurrence over a fixed time or space, which is similar to the Binomial Distribution's assumption of a fixed number of trials.

4. What is the main difference between the Poisson Distribution and the Binomial Distribution?

The main difference between the Poisson Distribution and the Binomial Distribution is that the Poisson Distribution models the number of events occurring in a fixed time or space, while the Binomial Distribution models the number of successes in a fixed number of trials. Additionally, the Poisson Distribution assumes a fixed rate of occurrence, while the Binomial Distribution assumes a fixed probability of success.

5. How can the Central Limit Theorem be used to analyze data from the Binomial Distribution?

The Central Limit Theorem can be used to analyze data from the Binomial Distribution by approximating the Binomial Distribution with a normal distribution. This allows for the use of statistical techniques that are based on the normal distribution, such as calculating confidence intervals and conducting hypothesis tests. The larger the sample size, the better the approximation will be and the more accurate the analysis will be.

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