Question

Subject: Limit theorem

1. The number of cars sold weekly at a dealership is a random variable
with expected value 16. Provide an upper limit on the probability that
a) Sales made in the next week exceed 18;

b) Suppose that a variation in the number of cars sold weekly is equal to 9.
Provide a lower limit for a probability of a close number of sales
week is between 10 and 22, inclusive.

Answer 1

Solution :

Let the random variable X represents number of cars sold weekly at a certain delarship .

Given that expected value (mean value) of the random variable X is 16

If the random variable X takes only nonnegative values then for any value a>0

(a) Then the probability that the next week sales exceed 18 is calculated as follows   

= 16/19

= 0.8421

(b) P(10<X<22) =

= P(- 2 < z < 2) =  P(z < 2) - P(z < - 2)

= 0.9772 - 0.0228

= 0.9544