Exercise 3.12: Cauchy Distribution

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Cauchy PDF

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

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.