Question

Let x represent the amount of frozen yogurt (in hundreds of gallons) sold by the G&T restaurant on any day during the summer. Storage limitations dictate that the maximum amount of frozen yogurt that can be kept at G&T on any given day is 250 gallons. Records of past sales indicate that the probability density function for x is approximated by y(x) = 0.32x for 0 ≤ x ≤ 2.5.

(a) What is the probability that on some summer day, G&T will sell less than 100 gallons of frozen yogurt?
(b) What is the mean number of gallons of frozen yogurt that G&T expects to sell on a summer day? (Round your answer to the nearest gallon) gal

Answer 1

a) The probability of selling less than 100 gallons (x≤1) is P=0.16.

b) The mean number of gallons is M=80 gallons.

Explanation:

The probability of selling x, in hundred of gallons, on any day during the summer is y(x)=0.32x, in a range for x from [0;2.5].

The probability of selling less than 100 gallons (x≤1) is then:

`P(x\leq1)=\int_0^(100)0.32x\cdot dx\\\\P(x\leq1)=0.16(x_b^2-x_a^2)\\\\P(x\leq1)=0.16(1^2-0^2)=0.16`

The mean number of gallons can be calculated as:

`M=\int_0^(2.5)(y(x)/x)\cdot dx=\int_0^(2.5)(0.32)\cdot dx\\\\M=0.32(x_b-x_a)=0.32(2.5-0)=0.32*2.5=0.8`