About the joint density function

In summary, the conversation discusses the concept of a joint density function and how it applies to the specific case where x and y have individual density functions in the same probability space, but no joint density function exists. The conversation also touches on the idea of a signed measure and integrating a bounded function over a set of measure 0.
  • #1
simpleeyelid
12
0
Could anyone help to give an example

where in the same proba space, x and y have each the density function, while the joint density function does not exist?

Thanks in advance,

Best regards
 
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  • #2
Yeah, that's easy. Let x be any r.v. with a pdf. Then, let y = 2*x. Thus, y also has a pdf, but the joint distribution is entirely concentrated on a set of measure 0 (in the x-y plane, that is), and so there is no joint pdf.
 
  • #3
First, thanks very much indeed..

but, yes, honestly, not easy for me, I have to think about it...

how to interpret this by saying the derivative of the signed measure does not exsit??
 
  • #4
could we say that, since y=2x is a line in R², we have nothing to integrate? sorry that I am so unintuitive...
 
  • #5
Basically, yeah. Since the joint distribution is only nonzero on a set of measure 0 (in the xy plane), you can't have a bounded function that integrates to 1 over it (i.e., a joint pdf).
 
  • #6
OK, thanks for this, really clear explanation, best wishes for you~
 

FAQ: About the joint density function

What is a joint density function?

A joint density function is a mathematical function that describes the probability of two or more random variables occurring simultaneously. It is used to model the joint behavior of multiple variables and can provide insights into the relationships between them.

How is a joint density function different from a probability distribution function?

A joint density function is similar to a probability distribution function in that it also describes the probability of outcomes for a given set of variables. However, a joint density function describes the probabilities for multiple variables, while a probability distribution function describes the probabilities for a single variable.

What are the key properties of a joint density function?

The key properties of a joint density function are that it is non-negative, the area under the function is equal to 1, and it can be integrated to obtain probabilities for specific ranges of values for the variables.

How is a joint density function used in statistical analysis?

A joint density function can be used to calculate the probability of specific outcomes for multiple variables, as well as to calculate expected values, variances, and covariances. It is also used in hypothesis testing and to model complex relationships between variables.

Can a joint density function be used for any type of data?

Yes, a joint density function can be used for any type of data, as long as the data can be described by multiple variables. It is commonly used in fields such as economics, finance, and engineering to model complex systems and relationships between variables.

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