A cauchy random variable X has the following pdf in which b

A cauchy random variable, X, has the following p.d.f. in which b is any positive real number, and a is any finite real number. Show that a Cauchy random variable has no average value.\\Show that the c.d.f. for a Cauchy random variable is

Solution

Given that the p.d.f. of Cauchy distribution is,

f(x) = (b / ) / b²+(x-a)²    (x is from -infinity to +infinity)

Where, b is any positive real number.

a is any finite real number.

Show that Cauchy distribution has no average value.

E(x) = x * f(x) dx   (x is from -infinity to +infinity)

=   x * (b / ) / b²+(x-a)² dx (x is from -infinity to +infinity)

This integral is undefined.

Therefore mean does not exist.

C.D.F. for Cauchy Random variable is,

F(x) = f(u)du (u is from -infinity to x)

= (b / ) / b²+(u-a)² du (u is from -infinity to x)

And we know that the result,

1 / (x² + a²) dx = 1 / a tan^-1(x/a)

Here x = (u-a) and a = b,

by putting these values in the following result,

= (b / ) * (1 / b) tan-1 [ (u-a/b) ] (u is from -infinity to x)

=  (b / ) * (1 / b) [ tan^-1 (x-a)/b ] - tan^-1 (-infinity) ]

=  (b / ) * (1 / b) [tan^-1 (x-a)/b ] - [ - /2]

= 1 / [/2 + tan^-1 (x-a)/b ]

= 1 /2 + 1 / tan^-1 {(x-a)/b}

Hence the proof.