Exploring the Characteristic Function of Joint PDFs: Tips and Techniques

In summary, the conversation revolved around finding a good source for learning about the characteristic function of a joint pdf and whether there are any rules for obtaining it. The conversation also touched upon the concept of independent random variables and the product rule for their joint characteristic function. Additionally, there was a discussion about a specific example involving a particle with a waiting time density and jump density. The conversation concluded with a recommendation for a book on characteristic functions.
  • #1
emptyset
3
0
Hi.
Does anyone know a good source for learning about the characteristic function of a joint pdf. Is there any nice rules for that? For example assume having a waiting time density and a jump density which are independent (easy things first). Is there an elegant way to get the characteristic function of the process if I have the two individual cf's?
Thanks
 
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  • #2
emptyset said:
Hi.
Does anyone know a good source for learning about the characteristic function of a joint pdf. Is there any nice rules for that? For example assume having a waiting time density and a jump density which are independent (easy things first). Is there an elegant way to get the characteristic function of the process if I have the two individual cf's?
Thanks

According to wikipedia for independent random variables the joint characteristic function is just the product of the characteristic function for each random variable.

http://en.wikipedia.org/wiki/Charac..._theory)#Basic_manipulations_of_distributions
 
  • #3
Thanks for your reply. This is true for a sequence of independent (and not necessarily identically distributed) random variables. However, I am looking for the joint pdf.
As an example, assume you have a particle sitting around at x' for a random time tau with the distribution y(t) and then jumping to x'' with the jump distribution z(x).
I am looking for a nice way of dealing with the characteristic funtion of the joint pdf v(x,t)=y(t)z(x)
given I now the charactersitic functions of y and z.
 
  • #4
A good book on characteristic functions is by Lukacs (called characteristic functions).

In your example if you have [itex]v(x,t)=y(t)z(x)[/itex], then the cf would be
[tex]\psi(\theta)=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\theta x t}z(x)y(t) dx dt =\int_{\mathbb{R}}\psi_z(t\theta)y(t) dt[/tex]
 
  • #5
thanks a lot for the information. that reference will be usefull.
 

1. What is a characteristic function?

A characteristic function is a mathematical function that describes the probability distribution of a random variable. It is a Fourier transform of the probability density function and provides information about the moments of the distribution.

2. How is a characteristic function different from a probability density function?

A characteristic function is a complex-valued function while a probability density function is a real-valued function. Additionally, the characteristic function describes the entire probability distribution, while a probability density function only describes the shape of the distribution.

3. What is the significance of the characteristic function in statistics?

The characteristic function is an important tool in statistics as it allows for the calculation of moments, such as mean and variance, of a distribution. It also enables the manipulation and combination of different distributions to create more complex distributions.

4. How is the characteristic function used in hypothesis testing?

In hypothesis testing, the characteristic function is used to calculate the test statistic, which is then compared to a critical value to determine the significance of the results. It is also used to generate confidence intervals and perform goodness-of-fit tests.

5. Can the characteristic function be used for any type of distribution?

Yes, the characteristic function can be used for any type of distribution, including discrete, continuous, and mixed distributions. However, it is most commonly used for continuous distributions.

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