Moment generating functions of continous probability distributions

Thread starter reen
Start date Jun 21, 2013
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Distributions Functions Moment Probability

In summary, a moment generating function (MGF) is a mathematical function that uniquely defines a probability distribution by the expected value of e^(tx), where t is a real number and x is a random variable. It can be used to calculate moments of a distribution, such as the mean and variance, without having to find the distribution's density function. The main difference between a moment generating function and a characteristic function is that the moment generating function only exists for distributions with finite moments, while the characteristic function exists for all distributions. A moment generating function cannot directly be used to find the probability of a specific outcome, but it can be used to calculate moments which can then be used for finding probabilities through other methods. The moments of a distribution can

Jun 21, 2013

reen

derive the MGF hence find their mean and variance
1 weibull distribution
2 pareto distribution
3 lognormal distribution

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Jun 21, 2013

HallsofIvy

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Okay, what have you done? What are the the formulas for those distributions and what are their "moments"?

Do you know the definition of "moment generating function"?

1. What is a moment generating function?

A moment generating function (MGF) is a mathematical function that uniquely defines a probability distribution. It is defined by the expected value of e^(tx), where t is a real number and x is a random variable. MGFs are useful because they can be used to calculate moments of a distribution, such as the mean and variance, without having to find the distribution's density function.

2. How is a moment generating function different from a characteristic function?

A moment generating function is defined as the expected value of e^(tx), while a characteristic function is defined as the expected value of e^(itx), where i is the imaginary unit. The main difference is that the moment generating function only exists for distributions with finite moments, while the characteristic function exists for all distributions. Additionally, the moment generating function is easier to work with mathematically, while the characteristic function is more useful for theoretical proofs.

3. Can a moment generating function be used to find the probability of a specific outcome?

No, a moment generating function cannot directly be used to find the probability of a specific outcome. It is used to calculate moments of a distribution, such as the mean and variance, which can then be used to find probabilities through other methods, such as the Central Limit Theorem or the Normal distribution.

4. How do you interpret the moments of a distribution using its moment generating function?

The moments of a distribution can be interpreted by taking derivatives of the moment generating function and evaluating them at t=0. The first derivative evaluated at t=0 gives the mean of the distribution, the second derivative gives the variance, the third derivative gives the skewness, and so on. This allows for a quick and easy way to calculate the moments of a distribution without having to find its density function.

5. Are there any limitations to using moment generating functions?

Yes, there are some limitations to using moment generating functions. They only exist for distributions with finite moments, so they cannot be used for distributions with infinite or undefined moments, such as the Cauchy distribution. Additionally, they may not always exist for certain distributions, or they may not be easy to calculate. In these cases, alternative methods may need to be used to find the moments of a distribution.