Milk is delivered to a chain of regional supermarkets once a week. If the weekly volume of sales in thousands of gallons is a random variable with probability density function f(x) = 7(1 – x)^6 for 0 < x < 1 0 otherwise How much milk must the chain of supermarkets order for each delivery so that the probability of the supply being exhausted in a given week is just 2%?

Answer:The chain of supermarkets must order 428.1 gallons of milk.Step-by-step explanation:Let's define the random variable X.X : ''Weekly volume of sales in thousands of gallons''X is a continuous random variable.The probability density function for X is [tex]f(x)=7(1-x)^{6}[/tex] when 0 < x < 1[tex]f(x)=0[/tex]   Otherwise.Let's denote as ''a'' to the quantity of milk for the question.We are looking for :[tex]P(X>a)=0.02[/tex][tex]P(X>a)=\int\limits^1_a {f(x)} \, dx[/tex][tex]P(X>a)=\int\limits^1_a {7(1-x)^{6}} \, dx[/tex][tex]P(X>a)=(1-a)^{7}[/tex][tex](1-a)^{7}=0.02[/tex][tex]a=1-\sqrt[7]{0.02}[/tex][tex]a=0.4281[/tex]a is in thousands of gallons, therefore the chain of supermarkets must order  428.1 gallons of milk in order to satisfy the question.