Proving Cauchy Density Function for Z = X+Y

In summary, by expanding and simplifying the hint provided, it can be shown that Z has a Cauchy density function.
  • #1
glacier302
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Homework Statement



Let X and Y be independent random variables each having the Cauchy density function f(x)=1/(∏(1+x2)), and let Z = X+Y. Show that Z also has a Cauchy density function.

Homework Equations



Density function for X and Y is f(x)=1/(∏(1+x2)) .
Convolution integral = ∫f(x)f(y-x)dx .

The Attempt at a Solution



My book gives the following hint, saying to "check it":

f(x)f(y-x) = (f(x)+f(y-x))/(∏(4+y2)) + 2/(∏y(4+y2))(xf(x)+(y-x)f(y-x)) .

Using this hint, I'm able to solve the rest of the problem, but I can't figure out how to prove that this hint is true.

Any help would be much appreciated : )
 
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  • #2


Just expand out the righthand side and simplify.
 

1. What is the Cauchy density function for Z = X+Y?

The Cauchy density function for Z = X+Y is a type of probability distribution that describes the likelihood of obtaining a certain sum of two independent random variables, X and Y. It is defined by the following equation: f(z) = (1/π) * (1 / (1 + (z^2))), where z is the sum of X and Y.

2. How is the Cauchy density function derived for Z = X+Y?

The Cauchy density function for Z = X+Y is derived by taking the convolution of the probability density functions of X and Y. This involves integrating the product of the two individual density functions over all possible values of X and Y. The resulting function is the Cauchy density function for Z = X+Y.

3. What are the properties of the Cauchy density function for Z = X+Y?

The Cauchy density function for Z = X+Y has several important properties, including symmetry, heavy tails, and lack of moments. It is symmetric around its mean, which is equal to 0. It also has heavy tails, meaning that it assigns a higher probability to extreme values compared to other probability distributions. Additionally, it does not have any finite moments, making it difficult to calculate certain statistical measures such as variance.

4. How is the Cauchy density function used in practical applications?

The Cauchy density function is commonly used in statistics and probability theory to model situations where extreme values are more likely to occur. It has applications in fields such as finance, physics, and engineering. One example is in the study of stock market returns, where the Cauchy distribution is used to model the occurrence of large price changes.

5. Are there any limitations to using the Cauchy density function for Z = X+Y?

Yes, there are some limitations to using the Cauchy density function for Z = X+Y. One major limitation is that it does not have any finite moments, which means that certain statistical measures, such as variance, cannot be calculated. Additionally, the Cauchy distribution is sensitive to outliers, which can affect the accuracy of any statistical analysis using this function. It is important to carefully consider the data and its characteristics before using the Cauchy density function in any practical application.

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