Consider a random sample $x_1,x_2,$dots,$x_n$ from a location scale

family with density function:

$f(x,,)=\frac{1}{}f(\frac{x-}{}),$

where $$ and $$ are the location and scale parameters, respectively

a$.$ Show that $R_1=\frac{S}{}$ is a pivotal quantity for inference involving

$,$ where $S^2$ is the sample variance.

b$.$ Show that $\frac{(\stackrel{}{x})-}{S}$ is a pivotal quantity for inference involving $,$

where $\stackrel{}{x}$ and $S^2$ are the sample mean and sample variance,

respectively.