Question

In: Math

Jobs arrive at the server is a Poisson random variable with a mean of 360 jobs...

Jobs arrive at the server is a Poisson random variable with a mean of 360 jobs per hour. Find:

The probability of at least 8 jobs arriving in the interval [8:00, 8:02] a.m.
The probability of at most 8 jobs arriving in the interval [9:15, 9:18] p.m.
Using the exponential distribution (and its relation to the Poisson) find the probability of no jobs arriving in the interval [7:00, 7:01] p.m.

Expert Solution

Jobs arrive at the server is a Poisson random variable with a mean of jobs per hour, that is at a rate per hour.

Let N(t) be the number of jobs that arrive at the server in a time interval t hours. We can say that N(t) has a Poisson distribution with parameter

The probability that n jobs arrive in period t is given by

a) The probability of at least 8 jobs arriving in the interval [8:00, 8:02] a.m. (that is during time period t=2/60 hours) is

ans: e probability of at least 8 jobs arriving in the interval [8:00, 8:02] a.m. is 0.9105

b) The probability of at most 8 jobs arriving in the interval [9:15, 9:18] p.m. (that is during time period t=3/60 hours) is

ans: The probability of at most 8 jobs arriving in the interval [9:15, 9:18] p.m is 0.0071

c. Let X be the inter arrival time (expressed in hours) between 2 jobs.

We can say that X has an exponential distribution with parameter

The pdf of X is

The Exponential cdf of X is

the probability of no jobs arriving in the interval [7:00, 7:01] p.m is same as the probability that the time taken for the arrival of the first job in the interval starting at 7:00 is greater than 1 minute (that is t=1/60 hours, t is expressed in hours), that is X>1/60

ans: the probability of no jobs arriving in the interval [7:00, 7:01] p.m. is 0.0025

milcah answered 2 months ago

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