The lodge owner also has a farm that raises free$-$range chickens that produce eggs. Eggs may be classified as small, if they weigh less than $44$ g $,$ medium, between $44$ g and $58$ g or large, weighing above $58$ g $.$ In a farm, egg weights follow a Normal distribution with a mean of $50$ g and a standard deviation of $5$ g $.$ The farm sells $1000$ eggs per day. $($a$)$ Find the proportions of small, medium and large eggs. $($b$)$ Find the expected value and the variance of the profit per day if the price of a small, medium and large egg are $10$ pence, $12$ pence and $15$ pence, respectively and there is a cost of $4$ pence to produce each egg.