Question

The daily amount of​ coffee, in​ liters, dispensed by a machine located in an airport lobby is a random variable X having a continuous uniform distribution with Aequals9 and Bequals12. Find the probability that on a given day the amount of coffee dispensed by this machine will be ​(a) at most 10.5 ​liters; ​(b) more than 9.4 liters but less than 11.2 ​liters; ​(c) at least 11.4 liters.

Answer 1

a) 1/2

b) 3/5

c) 1/5

Explanation:

`f(x)=(1)/((b-a)) , a\leq x\leq b; E(X)=(a+b)/(2)`

and
`Sigma^(2) =((b-a)^(2) )/(12)`

X ~ Uniform(9,12),

`f(x)=(1)/(b-a)=(1)/(12-9)=(a)/(3) , 9\leq x\leq 12`

a)
`p(x\leq 10.5)=\int\limits^a_b {(1)/(3) } \, dx =(10.5-9)/(3)=(1)/(2)`

Where b = 9 and a = 10.5

b)
`p(9.4\leq x\leq 11.2)=\int\limits^a_b {(1)/(3) } \, dx =(11.2-9.4)/(3)=(3)/(5)`

Where b = 9.4 and a = 11.2

c)
`p(x\geq  11.4)=\int\limits^a_b {(1)/(3) } \, dx =(12-11.4)/(3)=(1)/(5)`

Where b = 11.4 and a = 12