The time it takes a ski patroller to respond to an accident

The time it takes a ski patroller to respond to an accident call has an exponential distribution with an average equal to 5 minutes.

(a)  

In what time will 70 percent of all ski accident calls be responded to? (Round the intermediate value to 4 decimal places and then round your final answer to 2 decimal places.)

(b)  

If the ski patrol would like to be able to respond to 70 percent of the accident calls within 18 minutes, what does the average response time need to be? (Round the intermediate value to 4 decimal places and then round your final answer to 2 decimal places.)

Solution

a)

Here, the cumulative distribution, as the mean is 5 minutes, is

F(x) = 1 - exp(-(1/5)x)

Thus, if P(X<x) = 0.70, then

0.70 = 1 - exp(-(1/5)x)

0.30 = exp(-(1/5)x)

ln 0.30 = -(1/5)x

x = -5 ln (0.30) = 6.019864022 minutes [ANSWER]

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b)

Let u = the mean.

Here, the cumulative distribution, as the mean is u minutes, is

F(x) = 1 - exp(-(1/u)x)

Thus, if P(X<x) = 0.70, then

0.70 = 1 - exp(-(1/u)*18)

0.30 = exp(-18/u)

ln (0.30) = -18/u

u = -18 / ln(0.30) = 14.95050381 minutes [ANSWER]