A problem in understanding distributions exercise

In summary, a distribution refers to the way data is spread out and provides information about the frequency of different values. Understanding distributions is important for drawing conclusions from data. Common types include normal, skewed, and bimodal distributions. Distributions can be visualized using graphs like histograms and box plots. Factors like sample size, outliers, and data type can affect the shape of a distribution.
  • #1
Goldbeetle
210
1
I'm reading the first chapters of "A Guide to Distribution Theory and Fourier Transforms".
On page 10, Exercises 3,6,7 the distribution is defined in terms of integrals. The first one is always without integrand (there's only the integral sign). What does that mean? Am I missing something? The book can be consulted online in Google Books.

Thanks for any help!

Goldbeetle
 
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  • #2
I think it is just you intergrate over two disconnected ranges (-inf,-a) + (+a,+inf).
 
  • #3
Exactly

[tex]\int_{-\infty}^{-a}+\int_{a}^{\infty}\;f(x)dx=\int_{-\infty}^{-a}\;f(x)\;dx+\int_{a}^{\infty}\;f(x)\;dx[/tex]
 
  • #4
OK, thanks.
 
  • #5


Hi Goldbeetle,

It is common in distribution theory to define distributions in terms of integrals. This is because distributions are generalized functions that do not have a well-defined pointwise value, so they cannot be evaluated at a single point like regular functions can. Instead, distributions are defined by how they act on test functions, which are smooth functions with compact support. This is similar to how we define integrals in terms of limits of Riemann sums, rather than evaluating them at a single point.

In the exercises you mentioned, the integral without an integrand is likely meant to be interpreted as an integral over all of space, or an integral over the entire domain of the distribution. This is because distributions are defined over all of space, and the integral is used to represent how the distribution acts on test functions over its entire domain.

I hope this helps clarify the concept of distributions and their definition in terms of integrals. Keep up the good work studying distribution theory and Fourier transforms!
 

FAQ: A problem in understanding distributions exercise

1. What exactly is a distribution?

A distribution refers to the way in which a set of data is arranged or spread out. It provides information about the frequency of occurrence of different values in a dataset and helps to understand the overall pattern of the data.

2. Why is it important to understand distributions?

Understanding distributions is crucial for making sense of data and drawing meaningful conclusions. It allows scientists to identify trends, patterns, and outliers in the data, which can provide valuable insights and guide further research.

3. What are some common types of distributions?

Some common types of distributions include normal (or Gaussian) distribution, skewed distributions (such as right-skewed or left-skewed), and bimodal distributions. There are also discrete distributions, which are used for categorical or count data.

4. How can we visualize distributions?

Distributions can be visualized using various graphs and charts, such as histograms, box plots, and scatter plots. These visualizations help to understand the shape, center, and spread of the data.

5. What factors can affect the shape of a distribution?

The shape of a distribution can be affected by several factors, including the sample size, the presence of outliers, and the underlying population or process being studied. Additionally, the type of data (continuous or categorical) and the chosen bin size for a histogram can also impact the shape of a distribution.

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