The distance, X, between consecutive anomalies on long cable has an exponential distribution with mean 12 meters. Thus, the density of X is::

f(x,) =e^x

for 0x,=1

12

.is what R calls rate.

Hint: This is a problem involving the exponential distribution. Knowing the parameterfor the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts.

a) What is the expected value of X?

b) What is the variance of X?

c) What is the standard deviation of X?

d) What is the probability that X is larger than its expected value?

e) What is the probability that X is > 13?

f) What is the probability that X is > 14?

g) What is the probability that X > 14 given that X > 13?

h) What is the median of X?