Question

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 1

The chain of supermarkets must order 428.1 gallons of milk.

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

`f(x)=7(1-x)^(6)` when 0 < x < 1

`f(x)=0` Otherwise.

Let's denote as ''a'' to the quantity of milk for the question.

We are looking for :

`P(X>a)=0.02`

`P(X>a)=\int\limits^1_a {f(x)} \, dx`

`P(X>a)=\int\limits^1_a {7(1-x)^(6)} \, dx`

`P(X>a)=(1-a)^(7)`

`(1-a)^(7)=0.02`

`a=1-\sqrt[7]{0.02}`

`a=0.4281`

a is in thousands of gallons, therefore the chain of supermarkets must order

428.1 gallons of milk in order to satisfy the question.