Answer:
The answer is below
Step-by-step explanation:
The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes. (a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose that you have already been waiting for one hour for a taxi. What is the probability that one arrives within the next 10 minutes? (c) Determine x such that the probability that you wait more than x minutes is 0.10. (d) Determine x such that the probability that you wait less than x minutes is 0.90.
Answer:
A continuous variable is has a probability distribution function as:
[tex]f(x)=\lambda e^{-\lambda x}[/tex]
A continuous variable is has a cumulative distribution function as:
[tex]F(x)=1- e^{-\lambda x}[/tex]
Given mean = 10 minutes
[tex]mean=\frac{1}{\lambda}\\ \\10=\frac{1}{\lambda}\\\\\lambda=\frac{1}{10}[/tex]
a) P(X > 1 hour) = P(X > 60 minutes) = 1 - F(60) = [tex]1-[1-e^{-60*0.1}]=0.0025[/tex]
b) P(X < 70 | x > 60) = P(X < 10) = [tex]1-e^{-10*0.1}=0.6321[/tex]
c) 0.1 = P(X > x)
0.1 = 1 - F(x)
[tex]0.1=1-[1-e^{-0.1*x}\\\\0.1=e^{-0.1x}\\\\ln(0.1)=-0.1x\\\\-2.3=-0.1x\\\\x=23\ minutes[/tex]
d) 0.9 = P(X < x)
0.9 = F(x)
[tex]0.9=1-e^{-0.1*x}\\\\e^{-0.1x}=1-0.9\\\\e^{-0.1x}=0.1\\\\ln(0.1)=-0.1x\\\\-2.3=-0.1x\\\\x=23\ minutes[/tex]
