The Alfredo Fragrance Company produces only one product, a perfume called Hint of Elegance. Hint of Elegance consists of two secret ingredients blended into an exclusive fragrance, which is marketed in Zurich. An economic expression referred to as the Cobb$-$Douglas function describes the production of Hint of Elegance as follows:Where X is the amount of perfume produced.The company operates at a level where ingredient $1$ is set daily at $25$ units and ingredient $2$ at $36$ units. Although the price Alfredo pays for ingredient $1$ is fixed at $$50$ per unit, the cost of ingredient $2$ and the selling price for the final perfume are both probabilistic. The sales price for Hint of Elegance follows this distribution:SALES PRICE $($$$)$PROBABILITY$3000\mathrm{.}23500\mathrm{.}54000\mathrm{.}3$The cost for ingredient $2$ isIngredient $2$ Cost $($$$)$Probability$350\mathrm{.}1400\mathrm{.}6450\mathrm{.}3($a$)$ What is the profit equation for Alfredo Fragrance Company?$($b$)$ What is the expected profit to the firm?$($c$)$ Simulate the firm's profit for a period of nine days, using these random numbers from Table $14\mathrm{.}5$: $52,06,50,88,88,53,30,10,47,99,37,66,91,35,32,00,84,57,07.($d$)$ What is the expected daily profit as simulated in part $($c$)?[10$ Marks $]$