Explanation for moment-generating function

[paotrader]

Hello everyone, I have taken up on reading a mathematical statistics book and have gotten stuck on the moment-generating function. I tried using Wikipedia for a simpler explanation to no avail...
I noticed it is used a lot in finding the mean & variance for all types of distributions. Can someone explain to me in layman's terms the moment generating function. I can't seem to connect the dots...

Thanks in advanced for any help...Paolo

P.S. I had calculus courses about 20 years ago...

 

[ZioX]

The nth moment about the origin is defined as

$$E[X^n]=\int_{-\infty}^{+\infty}x^nf(x)dx$$

The mean is of course $E[x]=\mu$. The variance is $E[(X-E[X])^2]$ which can be shown to be equal to $E[X^2]-E[X]^2$.

The main point is that if you take the the nth derivative of the moment generating function and evaluate it at zero you get the nth moment about the origin. In symbols, $$E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n M_X(t)}{\mathrm{d}t^n}\right|_{t=0}$$

The proof of the continuous case is given on the wikipedia page (the notation on wikipedia slightly differs, m_i is the ith moment about the origin, so $m_i=E[X^i]$). The moment generating function can be used to calculate the mean as $M'_X(0)$ and the variance as $M''(0)-M'(0)^2$. You can look at the various wikipedia pages on particular distributions (like the poisson distribution) and calculate the means and variances from the moment generating function. If you're feeling adventurous you could calculate the moment generating function from the definition, $E[e^{tx}]$.

 

[tacman]

Hi Paolo,

The moment-generating function is a mathematical tool used in statistics to help us understand the behavior of a probability distribution. It is a function that takes in a variable, usually represented by the letter t, and produces a number that represents a specific moment of the distribution. These moments can include the mean, variance, skewness, and kurtosis of the distribution.

In simpler terms, the moment-generating function helps us to find important characteristics of a distribution, such as its average and how spread out its values are. It does this by using the properties of calculus, which is why you may remember it from your calculus courses.

To understand it better, think of the moment-generating function as a machine that takes in a distribution and spits out its key features. Just like how a car's speedometer shows us how fast it's going, the moment-generating function shows us important information about a distribution. It's a useful tool for statisticians and mathematicians, but it can also be helpful for anyone studying probability and data analysis.

I hope this helps you connect the dots and understand the moment-generating function better. If you have any further questions, feel free to ask. Good luck with your studies!