# Exercise 3.12: Cauchy Distribution

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The probability density function $\rm (PDF)$ of the Cauchy distribution is given as follows:

- $$f_x(x)=\frac{\rm 1}{\rm 2 \pi}\cdot \frac{\rm 1}{\rm 1+ (\it x/\rm 2)^{\rm 2}}.$$

From the graph you can already see the extremely slow decay of the PDF course.

Hints:

- The exercise belongs to the chapter "Further Distributions".
- In particular, reference is made to the section "Cauchy PDF".

### Questions

### Solution

**(1)**Comparing the given PDF with the general equation in the theory part, we see that the parameter is $\lambda= 2$.

- From this follows (after integration over the PDF):

- $$F_x ( r ) =\frac{1}{2} + \frac{\rm 1}{\rm \pi}\cdot \rm arctan(\it r/\rm 2).$$

- In particular.

- $$F_x ( r = +2 ) =\frac{1}{2} + \frac{\rm 1}{\rm \pi}\cdot \rm arctan(1)=\frac{1}{2} + \frac{\rm 1}{\rm \pi} \cdot \frac{\rm \pi}{4 }=0.75,$$
- $$F_x ( r = -2 ) =\frac{1}{2} + \frac{\rm 1}{\rm \pi}\cdot \rm arctan(-1)=\frac{1}{2} - \frac{\rm 1}{\rm \pi} \cdot \frac{\rm \pi}{4 }=0.25.$$

- The probability we are looking for is given by the difference:

- $${\rm Pr} (|x| < 2) = 0.75 - 0.25 \hspace{0.15cm}\underline{=50\%}.$$

**(2)** According to the result of the subtask **(1)** ⇒ $F_x ( r = 4 ) = 0.5 + 1/\pi = 0.852$.

- Thus, for the "complementary" probability: ${\rm Pr} (x > 4)= 0.148$.
- For symmetry reasons, the probability we are looking for is twice as large:

- $${\rm Pr} (|x| >4) \hspace{0.15cm}\underline{ = 29.6\%}.$$

**(3)** __All proposed solutions__ are true:

- For the variance of the Cauchy distribution holds namely:

- $$\sigma_x^{\rm 2}=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \hspace{-0.15cm} \frac{\it x^{\rm 2}}{\rm 1+(\it x/\rm 2)^{\rm 2}} \,\,{\rm d}x.$$

- For large $x$ the integrand yields the constant value $4$. Therefore the integral diverges.
- Chebyshev's inequality does not provide an evaluable bound, even with $\sigma_x \to \infty$.
- "Natural" random variables (physically interpretable) can never be cauchy distributed, otherwise they would have an infinite power.
- On the other hand, an "artificial" (or mathematical) random variable is not subject to this restriction. Example:
**The quotient of two zero mean quantities**.