Question

Show that the \(t\) -distribution with \(r=1\) degree of freedom and the Cauchy distribution are the same.

Short answer

The t-distribution with 1 degree of freedom and the Cauchy distribution are the same.

Step-by-step solution

Write down the formula for t-distribution

The probability density function of t-distribution is given by\n\[ p(x) = \frac{(1+x^{2}/r)^{- (r+1)/2}}{B(r/2, 1/2)}, \]\nwhere B is the beta function, x is the random variable, and r is the degree of freedom.

Write down the formula for Cauchy distribution

The probability density function of Cauchy distribution is given by\n\[ p(x) = \frac{1}{π(1+x^{2})},\] \nwhere x is the random variable.

Plug r = 1 into the t-distribution formula

In this case, t-distribution transforms into\n\[ p(x) = \frac{1}{B(1/2, 1/2)(1+x^{2})}, \] \nNote B(1/2, 1/2) = π.

Compare the two distributions

From step 3, as \(B(1/2, 1/2) = π\), we have\n\[ p(x) = \frac{1}{π(1+x^{2})}. \] \nWe can observe that, the two distributions ie., t-distribution (with r=1) and Cauchy distribution are identical.