$6.$ There are two boxes. In each box there are $4$ cards with a different number printed on it$.$ The four cards have been numbered as $1,2,3,4$ in each box. Two cards are drawn random from each box. The random variable X represents the difference between the number on the card from box $1$ minus the number on the card from box $2.$

a$)$ Find the mean of this probability distribution. $($i$.$e$.$ Find E$[$X$])$

b$)$ Find the variance and standard deviation of this probability distribution.

$($i$.$e$.$ Find V$[$X$]$ and SD$[$X$])$

The random variables M and W are defined as follows:

M $=$ X$+5$ and W $=(1/2)$X$+5$

c$)$ Find E$[$M$]$ and E$[$W$].$

d$)$ Find V$[$M$]$ and V$[$W$].$

e$)$ Mary and William play a game using the cards in the above boxes. Randomly two cards are drawn, and Mary records his score using the random variable M and William uses the random variable W$.$ They repeat this for a large number of times and compare their scores. Comment on any likely differences or similarities of their scores.