Discrete Random Vectors vs. Continuous Random Vectors

In summary, the conversation discusses the methods to check whether a given continuous or discrete random vector with a joint density function is indeed a joint density, with the use of integrals or summations respectively. The speaker also mentions the differences in the use of integrals and summations for finding expectations in discrete and continuous cases, and the importance of measure theory in making probability rigorous. They suggest that these topics can be further explored in a graduate or pure probability course, or a course on financial calculus.
  • #1
25
1
Given a continuous random vector (X,Y) with a joint density function
In order to check whether it is indeed a joint density ƒ(x,y) the method is to check if ∫∫ƒ(x,y)dxdy=1 where the integrals limits follow the bounds of x and y.

However, is it the case that if given an arbitrary discrete random vector (X,Y) with a joint density function:
In order to check whether it is indeed a joint density ƒ(x,y) the method is to check if ∑∑ƒ(x,y)=1 where the summations are, given bounds, for all those x and y.

Or are they both the ∫∫ f(x,y)dxdy=1? I know further on for finding the expectation and such that discrete uses the summation and continuous takes the integral but not sure if it differs in the first step or why it does.
 
Physics news on Phys.org
  • #2
The sum can be looked at as an integral using delta functions. Is that what you are aiming for?
 
  • #3
Technically the measure changes and the requirements for axiomatic probability are quite precise - and often difficult to follow.

If you are interested in this then you can find it in a graduate (or "pure") probability course that involves measure theory, sigma algebras, and the use of set theory to make probability rigorous over a variety of spaces (which as it turns out - is quite difficult).

A course on financial calculus (as a graduate offering) will cover this in one form or another.
 

What is the difference between a discrete random vector and a continuous random vector?

A discrete random vector is a collection of discrete random variables, where the values can only take on a finite or countably infinite number of values. A continuous random vector is a collection of continuous random variables, where the values can take on any real number within a given range.

How are the probabilities calculated for a discrete random vector and a continuous random vector?

For a discrete random vector, the probabilities are calculated by summing the individual probabilities of each possible combination of values. For a continuous random vector, the probabilities are calculated by integrating the probability density function over a given range of values.

What types of data can be represented by a discrete random vector and a continuous random vector?

A discrete random vector can represent data that can be counted, such as the number of heads in a series of coin flips. A continuous random vector can represent data that can be measured, such as the height or weight of individuals.

Can a discrete random vector be converted into a continuous random vector?

No, a discrete random vector and a continuous random vector represent fundamentally different types of data and cannot be converted into one another.

What are some examples of real-world applications of discrete random vectors and continuous random vectors?

Discrete random vectors can be used in applications such as modeling the outcomes of games or predicting customer preferences. Continuous random vectors can be used in applications such as weather forecasting or financial modeling.

Suggested for: Discrete Random Vectors vs. Continuous Random Vectors

Replies
25
Views
2K
Replies
2
Views
1K
Replies
9
Views
312
Replies
5
Views
238
Replies
30
Views
2K
Replies
10
Views
892
Back
Top