# Moment generating fuctions

1. Nov 8, 2015

### Josh S Thompson

mgfs can be used to find the moments of a random variable with relative ease.

But you have to evaluate the function at t=0. Many times these functions are such that t can be all real numbers.

How would you interpret an mgf evaluated at t=1. How does t relate to the probability distribution.

2. Nov 9, 2015

### andrewkirk

Since the mgf of X is $t\mapsto \mathbb{E}[e^{tX}]$, its evaluation at $t=1$ is $\mathbb{E}[e^{X}]$ which will, for instance, be of interest if you are working with exponential functions of the random variable $X$.

One relationship of mgf to the pdf is (apart from the obvious one that the mgf is obtained by integrating a function that involves the pdf), according to wiki, that:

'An important property of the moment-generating function is that if two distributions have the same moment-generating function [ie giving identical values for all values of $t$], then they are identical at almost all points.'

3. Nov 9, 2015

why

4. Nov 9, 2015

### Josh S Thompson

what do you need t for. I think if a two random variables have the same moments then they have the same distribution because the moments describe the spread of the data or the spread of the random variable and just thinking it out if you have infinite moments why does that not describe the random variable with infinite precision.

5. Nov 9, 2015

### Josh S Thompson

For example if you have [2,3,4,3,2,10,11] it would have a similar variance to say [2,5,6,3,7,3,2]
but it is different data, how do you use moments to see the difference in the data.

6. Nov 9, 2015

### andrewkirk

Those two things are sets of numbers, not random variables. Random variables have moment generating functions. Sets of numbers do not, without doing the additional work of building structure around them to turn them into a random variable.

The sets of numbers will have 'sample moments', but that's not enough to give a useful moment generating function.

7. Nov 9, 2015

### Josh S Thompson

ok so can you use sample moments to draw conclusions about that sample.

8. Nov 9, 2015

### Josh S Thompson

Can you explain how to interpret this;
E[x^3] - E[x^2]^(3/2)

9. Nov 9, 2015

### andrewkirk

One doesn't need to draw conclusions about the sample, because all the possible information about the sample is plainly visible right there in the sample. What one typically does is use the sample to draw conclusions (make estimates) about the population from which it has been sampled. The sample moments can be used for that. The simplest example of doing that is where the sample mean is used as an estimate of the population mean.

10. Nov 10, 2015

### Josh S Thompson

I understand that. This is what I'm trying to do; I'm looking at two samples of data [2,3,4,3,2,10,11] which has an average difference from the mean of 3.14 and [1,1,8,7,8,2,1] which also has an average difference from the mean of 3.14.

Surely these two data samples are not the same. If you calculate standard deviation the first set of numbers has a higher standard deviation because it incorporates the second moment. I'm asking if higher moments can be used to look at the spread of data and how you would do this. I think its just one of those things that you get or don't get. Please don't answer and talk about estimators for a population because in the real world things aren't distributed with two parameters.

11. Nov 10, 2015

### andrewkirk

Of course they're not the same. For a start, one has a lower minimum and the other has a higher maximum.

I'm afraid I can't see what this has to do with evaluating MGFs at t=1.

12. Nov 10, 2015

### Josh S Thompson

Why would you be working with exponential functions of a random variable

13. Nov 10, 2015

### andrewkirk

A very common example is the lognormal distribution, which is ubiquitous in finance. It's the distribution of a random variable that is the exponent of a normally distributed RV.

14. Nov 10, 2015

### Josh S Thompson

how do you derive that distribution

15. Nov 10, 2015

### Josh S Thompson

And what do you use it for. I'm pretty sure they only use it in academia for sum reason

16. Nov 10, 2015

### Josh S Thompson

Does it have a cdf or something because that's not that good of a reason but what do I know.

17. Nov 10, 2015

### andrewkirk

As I said it is used everywhere in finance - and in the business world, not just in academia. Pricing and valuation in insurance and banking would be unrecognisable without it. If you'd like to know more about it, the wikipedia article I linked above is very good.

It's used because it's skew and bounded below at zero, which is a feature of many random variables in finance like insurance claim sizes or investment returns; and also because it's closely related to the normal distribution, which is very well understood and easy to manipulate.

18. Nov 10, 2015

### Josh S Thompson

Ok thank you I appreciate the help but I still think you can use moments of a sample to determine the spread of the data