The number of major faults on a randomly chosen 1 km stretch of highway has a Poisson distribution with mean 1.3. The random variable $X$ is the distance (in km) between two successive major faults on the highway.

Part a) What is the probability of having at least one major fault in the next 2 km stretch on the highway? Give your answer to 3 decimal places.

Part b) Which of the following describes the distribution of $X$, the distance between two successive major faults on the highway?

A. $X \sim \mbox{Poisson}(2 \cdot 1.3)$
B. $X \sim \mbox{Exponential}( \mbox{mean} = 2 \cdot 1.3 )$
C. $X \sim \mbox{Exponential}( \mbox{mean} = \frac{1}{ 1.3 } )$
D. $X \sim \mbox{Poisson}(1.3)$
E. $X \sim \mbox{Exponential}( \mbox{mean} = \frac{1}{ 2 \cdot 1.3 } )$

Part c) What is the mean distance (in km) and standard deviation between successive major faults?

A. mean = 1.3; standard deviation = 1.3
B. mean = 0.7692; standard deviation = 0.5917
C. mean = 0.3846; standard deviation = 0.3846
D. mean = 0.7692; standard deviation = 0.7692
E. mean = 2.6000; standard deviation = 2.6000

Part d) What is the median distance (in km) between successive major faults? Give your answer to 2 decimal places.

Part e) What is the probability you must travel more than 3 km before encountering the next four major faults? Give your answer to 3 decimal places.

Part f) By expressing the problem as a sum of independent Exponential random variables and applying the Central Limit Theorem, find the approximate probability that you must travel more than 25 km before encountering the next 33 major faults? Give your answer to 3 decimal places. Please use R to obtain probabilities and keep at least 6 decimal places in intermediate steps.