Respuesta :
Answer:
(a) The probability that a person has to wait less than 6 minutes for the bus is 0.24.
(b) The probability that a person has to wait between 10 and 20 minutes for the bus is 0.40.
Step-by-step explanation:
Let The random variable X be defined as the waiting time for a bus at a certain bus stop.
The random variable X follows a continuous Uniform distribution with parameters a = 0 and b = 25.
The probability density function of X is:
[tex]f_{X}(x)=\left \{ {{\frac{1}{b-a};\ a<X<b;\ a<b} \atop {0;\ otherwise}} \right.[/tex]
(a)
Compute the probability that a person has to wait less than 6 minutes for the bus as follows:
[tex]P(X<6)=\int\limits^{6}_{0}{\frac{1}{25-0}}\, dx[/tex]
[tex]=\frac{1}{25}\times \int\limits^{6}_{0}{1}\, dx[/tex]
[tex]=\frac{1}{25}\times [x]^{6}_{0}[/tex]
[tex]=\frac{1}{25}\times [6-0][/tex]
[tex]=0.24[/tex]
Thus, the probability that a person has to wait less than 6 minutes for the bus is 0.24.
(b)
Compute the probability that a person has to wait between 10 and 20 minutes for the bus as follows:
[tex]P10<(X<20)=\int\limits^{20}_{10}{\frac{1}{25-0}}\, dx[/tex]
[tex]=\frac{1}{25}\times \int\limits^{20}_{10}{1}\, dx[/tex]
[tex]=\frac{1}{25}\times [x]^{20}_{10}[/tex]
[tex]=\frac{1}{25}\times [20-10][/tex]
[tex]=0.40[/tex]
Thus, the probability that a person has to wait between 10 and 20 minutes for the bus is 0.40.
