Moment Generating Function

In summary, the conversation discusses defining a random variable X with a probability density function f(x) and a moment-generating function M_x(t). It then considers defining a new random variable Y as -X, and asks for the moment-generating function M_y(t) for Y. The conversation notes that one approach is to first find the probability density function f(y) and then use it to find M_y(t), but wonders if there is an easier way since M_x(t) is already known. The conversation also discusses simplifying the expression for M_cX(t) and how it relates to M_x(t). Finally, one of the speakers shares their solution for the problem using the form of the function of x.
  • #1
zli034
107
0
Say r.v. X, we have pdf f(x) and mgf Mx(t) defined.

Then define Y=-X, y is negative x.

Can we get mgf of Y, i.e. My(t) and how?

I know I can go the way to get pdf f(y) first then My(t). I want to know if Mx(t) is already in my hands, it should be easier to get My(t) other than do f(y) first.

I thought about this whole afternoon. Just don't get it. Please help and thanks in advance.

xoxo
 
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  • #2
Remember that for any random variable

[tex]
M_x(t) = E[e^{tx}]
[/tex]

If you want the mgf for cX (any constant times X)

[tex]
M_{cX}(t) = E[e^{t(cx)}]
[/tex]

How can you simplify this expression, how does it relate to [tex] M_X(t)[/tex],
and how do both observations relate to your problem?
 
  • #3
I have found one way.

I have here x=log(1-x) and f(x) also known.

So we can see if I put x=log(1-x) into Mcx(t), it can be simplified easily. Here is my solution for this problem and it depends on the form of function of x.
 
  • #4
How does your second post relate to your first question?
 

1. What is a moment generating function?

A moment generating function (MGF) is a mathematical function that uniquely defines the probability distribution of a random variable. It provides a way to calculate the moments of a distribution, such as the mean, variance, and higher order moments, by taking derivatives of the MGF.

2. How is a moment generating function related to a probability distribution?

The MGF of a probability distribution is a mathematical representation of that distribution. It encodes all the information about the distribution, including its shape, location, and spread. By calculating the moments of a distribution using the MGF, we can fully characterize the distribution.

3. How is a moment generating function used in statistics?

The MGF is a useful tool in statistics for several reasons. It allows us to calculate the moments of a distribution, which can be used to estimate unknown parameters or test hypotheses. It also enables us to find the distribution of a sum of independent random variables, which is often used in statistical modeling and analysis.

4. What are the properties of a moment generating function?

There are several important properties of a moment generating function, including:

  • The MGF exists for all values of t within a certain range.
  • The MGF uniquely defines the distribution of a random variable.
  • The MGF of a sum of independent random variables is the product of their individual MGFs.
  • The MGF of a linear combination of random variables is the product of their individual MGFs.

5. What is the difference between a moment generating function and a characteristic function?

Both moment generating functions and characteristic functions are used to represent probability distributions. However, the MGF is defined as a function of the real variable t, while the characteristic function is defined as a function of the imaginary variable i*t. Additionally, the MGF is useful for calculating moments, while the characteristic function is useful for calculating Fourier transforms.

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