MHB Moment-Generating Function question

  • Thread starter Thread starter JGalway
  • Start date Start date
  • Tags Tags
    Function
AI Thread Summary
The moment-generating function \( M_x(t) = \frac{t}{t-1} \) cannot correspond to any random variable because \( M_x(0) \) does not equal 1, which is a requirement for valid moment-generating functions. Differentiating the function yields a mean of 1 and a variance of 1, which initially appears acceptable. However, the failure of \( M_x(0) \) to equal 1 indicates that it does not satisfy the properties of a moment-generating function. This inconsistency confirms that no random variable can be represented by this moment-generating function. Understanding these properties is crucial for identifying valid moment-generating functions in probability theory.
JGalway
Messages
6
Reaction score
0
Q:Explain why there can be no random variable for which $M_x(t)=\frac{t}{t-1}$
($M_x(t)$ is the moment-generating function of the random variable $x$.)

A: I tried differentiating it twice and got mean of $x=1$ and variance of $x=1$ which seems fine. Maybe its because $M_x(0)$ is not equal to $1$?Thanks in advance for any help with this problem.
 
Physics news on Phys.org
JGalway said:
Maybe its because $M_x(0)$ is not equal to $1$?

Yes, that's exactly the reason.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
View full post »

Similar threads

Proving variance with moment generating functions
Replies
6
Views
2K
Chernoff Bounds using importance sampling identity
Replies
0
Views
2K
Calculating PDF from MGF: Advice Needed
Replies
6
Views
10K
I Does the generalized gamma distribution have a mgf?
Replies
1
Views
2K
I Understanding extinction probability in simple GWP
Replies
1
Views
2K
MHB Counterfactual Expectation Calculation
Replies
1
Views
2K
I How to obtain moment bound from the importance sampling identity?
Replies
1
Views
2K
Statistics - Moment Generating Functions
Replies
2
Views
1K
I What Are the Axioms of Fuzzy Logic and How Do They Extend Boolean Algebra?
Replies
7
Views
1K
I An Alternative Variance Formula
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top