Support of Stochastic Variables: Math_Ninja

[Math_Ninja25]

Homework Statement

Hi

I have been working on understanding concept fra Measure Theory known as support or supp

I know that according to the definition

if $$(\mathcal{X},\mathcal{T})$$ is a topological space and $$(\mathcal{X},\mathcal{T}, \mu)$$ such that the sigma Algebra A contain all open sets $$U \in \mathcal{T}$$. Then the support on a measure $$\mu$$ is defined as the set of all points $$x \in \mathcal{X}$$ for which every open neighbourhood of x has a positve measure

$$\mathrm{supp}(\mu) = \{x \in X| \forall z \in Z \in N_{x} \in \mathcal{T}, \mu(N_{x}) > 0 \}$$

Can this definition be used on a discrete stohastic vector $$(Y,Z)$$ with the distributionfunction $$p_{Y,Z}$$

$$P(Y = y, Z = z) = p_{Y,Z}(Y,Z) = \left\{ \begin{array}{ll}\frac{1}{2} \cdot e^{-\lambda} \frac{\lambda^z}{z!} & \mbox{where \ \mathrm{y \in \{-1,0,1\}} \mathrm{and} \mathrm{z \in \{0,1,\ldots\}}} \\ 0 & \mbox{\mathrm{other.}}\end{array} }\right$$

where $$\lambda > 0$$

I need to determine supp $$P_{Y,Z}$$

Then by the definition of supp then the non-negative mesure on a meassurable space (Y,Z), is the function

$$P: \rightarrow ]-1,1[ \ \mathrm{x} \ ]0,\infty [$$

Homework Equations

The supp is is it then the set of all subsets which the measure operates on?

The Attempt at a Solution

Therefore the support $$supp \ p_{(Y,Z)} = supp(P) = \{y \in Y| \forall z \in Z , P(Y,Z) > 0\}$$

Sincerely Yours
Math_Ninja

 

[mighty2000]

Dear Math_Ninja,

Thank you for your post. It seems like you are trying to understand the concept of support in measure theory and how it applies to a discrete stochastic vector with a given distribution function. Let me try to provide some clarification and help you with your question.

Firstly, the definition of support in measure theory is slightly different from the one you have provided. The correct definition is as follows:

Given a measurable space (X, Σ) and a measure μ on it, the support of μ is defined as the smallest closed set E such that μ(E^c) = 0, where E^c denotes the complement of E. In other words, the support of a measure is the set of all points where the measure is non-zero.

Now, to answer your question about whether this definition can be used on a discrete stochastic vector (Y, Z) with the given distribution function, the answer is yes. In this case, the measurable space would be the Cartesian product of the spaces of Y and Z, i.e. (Y x Z, Σ), where Σ is the sigma-algebra generated by the product of the sigma-algebras of Y and Z. The measure μ would be the distribution function p_{Y,Z}.

To determine the support of this measure, you would need to find the smallest closed set E such that p_{Y,Z}(E^c) = 0. In this case, the support would be the set of all (y,z) such that p_{Y,Z}(y,z) > 0. In other words, the support would be the set of all possible outcomes of the discrete stochastic vector (Y,Z) with non-zero probability.

I hope this helps. Let me know if you have any further questions or need any additional clarification.