1. 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

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Answer:

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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]

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