# Exercise 3.12: Cauchy Distribution

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:

### Questions

1

What is the cumulative distribution function  $\rm (CDF)$  $F_x(r)$?  What is the probability that  $|x|<2$?

 ${\rm Pr} (|x| < 2) \ = \$ $\ \%$

2

What is the probability that  $|x|>4$?

 ${\rm Pr} (|x| > 4) \ = \$ $\ \%$

3

Which of the following statements are true for the Cauchy distribution?

 The Cauchy distribution has an infinitely large variance. The Chebyshev inequality makes no sense here. A random variable that can be measured in nature is never Cauchy distributed.

### Solution

#### 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.