Thetime between calls is exponentially distributed with a mean timebetween calls

Question

Thetime between calls is exponentially distributed with a mean timebetween calls of 10 minutes.

a. What is theprobability that the time until the first call isless than 5 minutes?

b. What is theprobability that the time until the first call isbetween 5 and 15 minutes?

c. Determine the lengthof an interval of time such that the probability of atleast one call in the interval is 0.90.

d. If there has not beena call in 10 minutes, what is the probability that the time untilthe next call is more than 5 minutes?

Explanation / Answer

Let T1 =time of the 1st call.

T1 ~ Exponential( = 1/10 =0.1)

T ~ Exponential()

a) P(T1 < 5)

P(T < t) = 1 –e–t

P(T1 < 5) = 1 –e–(0.1)(5)= 1 –e–0.5

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