Need Help?

Subscribe to Probability

Subscribe
  • Notes
  • Comments & Questions

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

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

No comments yet

Join the conversation

Join Notatee Today!