Moment Generating Function problem

In summary, the conversation is discussing the use of the probability density function and moment generating function to find E[X^2]. While the integral definition gives a simple answer of 3/5, the use of the moment generating function results in an unbounded answer. It is noted that in some cases, the moment generating function may not be defined, as seen in the example of the uniform distribution.
  • #1
Glass
24
0
I'm given the probability density function:
f(x) = 3x^2 for x in [0, 1]
f(x) = 0 elsewhere
I want to find E[X^2] which is easy if I use the integral definition (I got 3/5). Yet, when I try and do this using Moment Generating Function (mgf) I cannot seem to get the same answer (in fact I get an unbounded answer). I'm wondering why this is the case. Any help?
 
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  • #2
Well as you guessed, that's not normal. So there has to be something quite not right with one of your calculations. But nobody can help you find what is the problem if we can't see your calculations.
 
  • #3
From the definition of mgf I have:
[tex] M(t) = \int_{0}^{1}e^{xt}3x^2dx [/tex]
Then after some integration by parts (or using a symbolic integrator) I get:
[tex] M(t) = e^t(\frac{3}{t} - \frac{6}{t^2} + \frac{6}{t^3}) - \frac{6}{t^3} [/tex]
Then if we differentiate that twice (or even once or zero times) there are still 1/t (or some power of 1/t) terms in there.
 
  • #4
Actually, I did the calculations, and I get what you're saying when you say it's unbounded. At first I imagined that by "unbounded" you meant that the integral blew up, because you forgot that the bounds were 0 and 1 and left them to ±infinity or something. But it's actually that M(0) is not defined.

This is nothing to be so surprised about though. There are many cases when you can't use the mgf because it simply is not defined on any nbhd of 0.

A trivial exemple of when the mgf does not exist is for the uniform distribution. We then have

[tex]M_X(t)=\int_{0}^{1}e^{tx}dx=\frac{1}{t}(e^t-1)[/tex]

Undefined at t=0.
 

What is a moment generating function?

A moment generating function (MGF) is a mathematical function used to describe the statistical properties of a probability distribution. It is defined as the expected value of e^(tx), where t is a variable and x is a random variable.

What is the purpose of using a moment generating function?

MGFs are used to find the moments of a probability distribution, such as the mean, variance, and higher order moments. They can also be used to prove theorems and simplify calculations in probability and statistics.

How do you find the moment generating function of a specific distribution?

The moment generating function of a specific distribution can be found by taking the expected value of e^(tx) where x is a random variable following that distribution. For example, the MGF of a normal distribution with mean mu and variance sigma^2 is e^(mu*t + (sigma^2*t^2)/2).

What is the relationship between the moment generating function and the characteristic function?

The characteristic function and the moment generating function are closely related. They both provide information about the statistical properties of a probability distribution. The characteristic function is the Fourier transform of the probability density function, while the MGF is the Laplace transform. Both can be used to find the moments of a distribution, but the MGF is easier to work with mathematically.

How can the moment generating function be used to find the distribution of a random variable?

If two random variables have the same moment generating function, they must have the same distribution. Therefore, by comparing the MGFs of two random variables, we can determine if they follow the same distribution. Additionally, the MGF can be used to find the distribution of a random variable by taking the inverse Laplace transform of its MGF.

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