5.6 The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that is 4 miles long. Experience has shown that the distance along this stretch, measured in miles from its northernmost point, at which an emergency occurs can be represented by a uniformly distributed random variable over the range 0 to 4 miles. Then, if X denotes the distance (in miles)

of an emergency from the northernmost point of this stretch of river, its probability density function is as follows:

f1x2 = e 0.25 for 0 6 x 6 4 0 for all other x

a. Graph the probability density function.

b. Find and graph the cumulative distribution function.

c. Find the probability that a given emergency arises within 1 mile of the northernmost point of this stretch of river.

d. The rescue team’s base is at the midpoint of this stretch of river. Find the probability that a given emergency arises more than 1.5 miles from this base.