A Particular Brand of Dishwasher Soap Is Sold in Three $X _ { 1 } \text { and } X _ { 2 }$

A particular brand of dishwasher soap is sold in three sizes: 25oz, 40oz, and 65 oz. Twenty percent of all purchasers select a 25 oz box, fifty percent select a 40 oz box, and the remaining thirty percent choose a 65 oz box. Let $X _ { 1 } \text { and } X _ { 2 }$ denote the package sizes selected by two independently selected purchasers.
a. Determine the sampling distribution of $\bar { X }$
, calculate $E ( \bar { X } )$
, and compare to $\mu .$
b. Determine the sampling distribution of the sample variance $S ^ { 2 } \text {, calculate } E \left( S ^ { 2 } \right) \text {, and compare to } \sigma ^ { 2 } \text {. }$