Sum of IID random variables and MGF of normal distribution

In summary, the conversation discusses whether the moment generating function (MGF) of all random variables raised to the Nth power will tend to the MGF of the normal distribution if the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity. The possibility of this and the proof steps are also questioned and the need for specifying the type of convergence is mentioned. The conclusion is that it is the mean of the sum, not the sum itself, that converges to a normal distribution.
  • #1
Luna=Luna
16
0
If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity, shouldn't the MGF of all random variables raised to the Nth power tend to the MGF of the normal distribution?

I tried to do this with the sum of bernouli variables and exponential variables and didn'treally get anywhere with either.

Does anyone know if this is even possible and where I can find the proof steps?
 
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  • #2
Luna=Luna said:
If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity

You have to specify what type of convergence you're talking about when you say "tends". ( http://en.wikipedia.org/wiki/Convergence_of_random_variables)

The sum of iid random variables doesn't converge (in distribution) to a normal distribution. It's the mean of the sum that converges to a normal distribution.
 

Related to Sum of IID random variables and MGF of normal distribution

1. What is the sum of IID random variables?

The sum of IID (independent and identically distributed) random variables is the total of all the individual random variables in a sample, where each variable is independent of the others and has the same probability distribution. This sum is also known as the sample mean or average.

2. How is the sum of IID random variables related to the central limit theorem?

The central limit theorem states that the sum of a large number of IID random variables will follow a normal distribution, regardless of the underlying distribution of the individual variables. This means that as the sample size increases, the sum of IID random variables will approach a normal distribution, making it a useful tool in statistical analysis.

3. What is the moment generating function (MGF) of a normal distribution?

The moment generating function (MGF) of a normal distribution is a mathematical function that uniquely describes the shape of a normal distribution. It is defined as the expected value of e^(tX), where t is a real number and X is a random variable following a normal distribution. The MGF of a normal distribution is e^(μt + σ^2t^2/2), where μ is the mean and σ^2 is the variance of the distribution.

4. How is the MGF of a normal distribution used to calculate probabilities?

The MGF of a normal distribution can be used to calculate probabilities by using the properties of the moment generating function. Specifically, the MGF can be used to find the moments (mean, variance, etc.) of a normal distribution, which can then be used to calculate probabilities using standard normal tables or statistical software.

5. What is the relationship between the MGF of a normal distribution and the sum of IID random variables?

The MGF of a normal distribution can be used to find the MGF of the sum of IID random variables. This is because the MGF of the sum of IID random variables is the product of the MGFs of each individual variable. Since the MGF of a normal distribution is known, this relationship can be used to find the MGF of the sum of IID random variables, and in turn, calculate probabilities and other statistical measures.

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