Proof of Normal approximation to Poisson.

In summary, the Poisson distribution tends to a Normal distribution for a large parameter lambda. This can be proven using Stirling's approximation and the Maclaurin series. By defining a new variable and assuming it to be much smaller than the mean, the Poisson distribution can be transformed into a normal distribution with the same mean and variance.
  • #1
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I have been looking for a proof of the fact that for a large parameter lambda, the Poisson distribution tends to a Normal distribution. I know the classic proof using the Central Limit Theorem, but I need a simpler one using just limits and the corresponding probability density functions. I was told this was really easy using Stirling's approximation:

n! ~ sqrt(2*pi*n) * (n/e)^n

but I just don't see it. Anyone knows this proof?
 
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  • #2
First you take the natural logarithm to the Poisson distribution and then apply Stirlings approximation. Then define a new variable

[tex]y=x-\mu[/tex]

and assume that y is much smaller than [tex]\mu[/tex]
By doing this you will end up with a term

[tex]\ln\left(1+\frac{y}{\mu}\right) [/tex]

which can be approximated by looking at the Maclaurin series

[tex]\ln\left(1+\frac{y}{\mu}\right) \approx \frac{y}{\mu} - \frac{y^{2}}{2\mu^{2}}. [/tex]

Now any term with a power of [tex]\mu[/tex] greater than 2 in the denominator may be approximated as zero due to the assumption that y is much smaller than [tex]\mu[/tex]. When the algebra is done you just takt the exponential function on both sides and you end up with a normal distribution with mean and variance [tex]\mu[/tex].
 

Related to Proof of Normal approximation to Poisson.

1. What is the definition of Normal approximation to Poisson?

The Normal approximation to Poisson is a statistical method used to approximate a Poisson distribution with a Normal distribution. It is based on the idea that as the sample size increases, the shape of the Poisson distribution becomes more similar to a Normal distribution.

2. How is Normal approximation to Poisson used in real-world applications?

Normal approximation to Poisson is commonly used in quality control and reliability testing, as well as in financial and insurance industries for modeling rare events. It is also used in genetics and epidemiology to analyze rare occurrences.

3. What are the assumptions of Normal approximation to Poisson?

The main assumptions of Normal approximation to Poisson are that the sample size is large enough and the probability of an event occurring is small. Additionally, the events must be independent and the mean and variance of the Poisson distribution must be equal.

4. How is Normal approximation to Poisson different from the Central Limit Theorem?

Normal approximation to Poisson is a specific application of the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means will approach a Normal distribution regardless of the underlying distribution of the population. Normal approximation to Poisson, on the other hand, specifically applies to approximating a Poisson distribution with a Normal distribution.

5. Are there any limitations to using Normal approximation to Poisson?

Yes, Normal approximation to Poisson is only accurate when the sample size is large and the probability of an event occurring is small. If these conditions are not met, the approximation may not be reliable. Additionally, it may not be appropriate to use this method for discrete data or in cases where the mean and variance of the Poisson distribution are not equal.

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