Poisson Distribution Homework: Use Central Limit Theorem

In summary, as the sample size increases, the sum of n Poisson distributed random variables with mean lambda/n approaches a normal distribution with mean lambda and variance lambda, which is the same distribution as the original Poisson distribution X with mean lambda. This can be determined using the Central Limit Theorem.
  • #1
rhyno89
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Homework Statement


We can approximate a poisson distribution from the normal. Suppose lambda is a large positive value; let X ~ Poisson(lambda) and let X1...Xn be independant identicly distributed from a Poisson (lambda/n) distribution. Then X and X1+...+Xn have the same distribution. Use the central limit theorem to determine the approximate distribution of X...


Homework Equations





The Attempt at a Solution



I know that the above statement is true through moment generating functions. If X is distributed with mean of lambda then n random variables with a mean of lambda over n will sum to a distribution with mean of lambda.

It seems from the question that the distribution of X would just be a poisson distribution with mean of lambda but that doesn't really require any work as the problem states that.

Any help on getting started?
 
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  • #2


I can confirm that the statement is true. This is because as the sample size (n) increases, the distribution of the sum of n independent and identically distributed random variables approaches a normal distribution, as stated by the Central Limit Theorem. In this case, the sum of n Poisson distributed random variables with mean lambda/n will approach a normal distribution with mean n(lambda/n) = lambda and variance n(lambda/n) = lambda. This is the same distribution as X, which is also Poisson with mean lambda. Therefore, we can approximate a Poisson distribution from the normal distribution.
 

FAQ: Poisson Distribution Homework: Use Central Limit Theorem

1. What is the Poisson distribution?

The Poisson distribution is a probability distribution that is often used to model the number of events that occur in a fixed interval of time or space. It is characterized by a single parameter, λ, which represents the average number of events that occur in the given interval.

2. How is the Poisson distribution related to the Central Limit Theorem?

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution. This means that for large enough sample sizes, the mean of a sample from a Poisson distribution will follow a normal distribution.

3. What is the formula for the Poisson distribution?

The formula for the Poisson distribution is P(x) = (e^-λ * λ^x) / x!, where x is the number of events, λ is the average number of events, e is the mathematical constant e, and x! is the factorial of x.

4. How do you use the Central Limit Theorem to solve Poisson distribution homework problems?

To solve Poisson distribution homework problems using the Central Limit Theorem, you first need to find the mean and standard deviation of the Poisson distribution. Then, you can use these values to calculate the z-score, and from there you can use a z-table or a calculator to find the probability or percentile that you need.

5. Can the Central Limit Theorem be applied to any sample size?

No, the Central Limit Theorem is only applicable to large sample sizes (typically n ≥ 30). For smaller sample sizes, other methods such as the t-distribution may need to be used to approximate the sampling distribution of the mean.

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