Chapter 5: Q.5.16 (page 218)

A standard Cauchy random variable has density function
$f (x) = \frac{1}{π (1 + x^{2})} - \infty < x < \infty$
Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.

Short Answer

Expert verified

$- \int_{- \infty}^{\infty} \frac{1}{π (1 + y^{2})} d y$ is also the probability density function of a standard Cauchy's distribution. So, $\frac{1}{x}$ is also standard Cauchy's distribution.

Step by step solution

Given information

A standard Cauchy random variable has density function

$f (x) = \frac{1}{π (1 + x^{2})} - \infty < x < \infty$

Solution

The probability density function of Cauchy's distribution will be,

$f (x) = \frac{1}{π (1 + x^{2})}, - \infty < x < \infty$

Now let's calculate,

$y = \frac{1}{x} \Rightarrow x = \frac{1}{y}$

We need to Differentiate on both sides

$d x = - \frac{1}{y^{2}} d y$

The probability density function of x is,

$f (x) = \int_{- \infty}^{\infty} \frac{1}{π (1 + x^{2})} d x$

$= \int_{- \infty}^{\infty} \frac{1}{π (1 + {(\frac{1}{y})}^{2})} - \frac{d y}{y^{2}}$

$= \int_{- \infty}^{\infty} \frac{1}{π (1 + \frac{1}{y^{2}})} - \frac{d y}{y^{2}}$

$= \int_{- \infty}^{\infty} \frac{y^{2}}{π (1 + y^{2})} - \frac{d y}{y^{2}}$

$= - \int_{- \infty}^{\infty} \frac{1}{π (1 + y^{2})} d y$

Final answer

$- \int_{- \infty}^{\infty} \frac{1}{π (1 + y^{2})} d y$ is also the probability density function of a standard Cauchy's distribution. So, $\frac{1}{x}$ is also standard Cauchy's distribution.

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A standard Cauchy random variable has density function
$f (x) = \frac{1}{π (1 + x^{2})} - \infty < x < \infty$
Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.

Short Answer

Step by step solution

Given information

Solution

Final answer

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A standard Cauchy random variable has density functionf(x)=1π1+x2−∞<x<∞Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.

Short Answer

Step by step solution

Given information

Solution

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

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A standard Cauchy random variable has density function
$f (x) = \frac{1}{π (1 + x^{2})} - \infty < x < \infty$
Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable.