3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus, the possible values of T are real numbers in the interval [0,1]) (a) What is P (0 T< t) forte [0,1]? Note that we are conditioning on exactly one occurrence/arrival during the first hour. Hint: Consider the number of occurrences/arrivals in the interval [0, t] and in the interval (t, 1], and then it becomes similar to Q2 above. (b) What is the cumulative distribution then of T under the same conditions?
3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus, the possible values of T are real numbers in the interval [0,1]) (a) What is P (0 T< t) forte [0,1]? Note that we are conditioning on exactly one occurrence/arrival during the first hour. Hint: Consider the number of occurrences/arrivals in the interval [0, t] and in the interval (t, 1], and then it becomes similar to Q2 above. (b) What is the cumulative distribution then of T under the same conditions?
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![3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence
occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus,
the possible values of T are real numbers in the interval [0,1])
(a) What is P (0 T< t) forte [0,1]? Note that we are conditioning on exactly one occurrence/arrival during
the first hour.
Hint: Consider the number of occurrences/arrivals in the interval [0, t] and in the interval (t, 1], and then it
becomes similar to Q2 above.
(b) What is the cumulative distribution then of T under the same conditions?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27e31058-a1cb-4ad6-9c59-ea40849f1a53%2F699348fe-75e3-4f78-a441-7af1f4157ed1%2Flb6ryga.png&w=3840&q=75)
Transcribed Image Text:3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence
occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus,
the possible values of T are real numbers in the interval [0,1])
(a) What is P (0 T< t) forte [0,1]? Note that we are conditioning on exactly one occurrence/arrival during
the first hour.
Hint: Consider the number of occurrences/arrivals in the interval [0, t] and in the interval (t, 1], and then it
becomes similar to Q2 above.
(b) What is the cumulative distribution then of T under the same conditions?
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