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• Notes

The time between arrivals of taxis at a busy intersec-
tion 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 you have already been waiting for one hour for a
taxi, what is the probability that one arrives within the
next 10 minutes?

$x \rightarrow$ exponential distribution

$\mu=10 \mathrm{min}$

(a) $P(x>60)$

$\mu=\frac{1}{\lambda} \rightarrow \lambda=\frac{1}{\mu}=0.1$

$F(x)=\lambda e^{-\lambda x}$

$P(x>60)=\int_{60}^{\infty} 0.1 e^{-0.1 x} d x=-\left.e^{-0.1}\right|_{60} ^{\infty}$

$=e^{-6}=0.0025$

(b) $P\left(x<60+10 | x>60\right)$

$=\int_{0}^{10} 0.1 e^{-0.1 x} d x=-e^{-0.1x}|_{0}^{10}$

$=1-e^{-1}=0.6321$

The distance between major cracks in a highway fol-
lows an exponential distribution with a mean of 5 miles.
(a) What is the probability that there are no major cracks in a
10-mile stretch of the highway?

(b) What is the probability that there are two major cracks in
a 10 -mile stretch of the highway?
(c) What is the standard deviation of the distance between
major cracks?

$x \rightarrow$ distance between cracks

$x \rightarrow$ exponential distribution

$\mu=5 \text { miles }$

(a) $P(x>10)$

$\lambda=\frac{1}{\mu}=\frac{1}{5}=0.2$

$P(x>10)=\int_{10}^{\infty} \lambda e^{-\lambda x} d x$

$=\int_{10}^{\infty} 0.2 e^{-0.2 x} d x=-\left.e^{-0.2 x}\right|_{10} ^{\infty}$

$=e^{-2}=0.1353$

(b) $y \rightarrow$ number of cracks in 10 miles

$P(y=2)=\frac{e^{-\lambda} \lambda^{y}}{y !}$

$\lambda=0.2 \times 10=2$

$P(y=2)=\frac{e^{-2} 2^{2}}{2 !}=0.2707$

(c) $\sigma_{x} \Rightarrow \sigma_{x}^{2}=\frac{1}{\lambda_{x}^{2}}$

$\sigma=\frac{1}{\lambda}=\frac{1}{0.2}=5$