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

Answer:

a) 1/2

b) 3/5

c) 1/5

Step-by-step explanation:

$f(x)=\frac{1}{(b-a)} , a\leq x\leq b; E(X)=\frac{a+b}{2}$

and $Sigma^{2} =\frac{(b-a)^{2} }{12}$

X ~ Uniform(9,12),

$f(x)=\frac{1}{b-a}=\frac{1}{12-9}=\frac{a}{3} , 9\leq x\leq 12$

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

Where b = 9  and  a = 10.5

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

Where b = 9.4  and  a = 11.2

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

Where b = 11.4  and  a = 12