Student ID Suppose that you receive hours phone calls with Poison rmals w a) Wha
ID: 3349826 • Letter: S
Question
Student ID Suppose that you receive hours phone calls with Poison rmals w a) What is the pdf for the time between two cahet is the iou don't receive an thr rupi-hon" call during the next two hours? ISP b) Say you did not receive any call during the last 7 hours so what is the probubility thur you don't receive any call during the next two hours? c) what is the expected time between your 4* and 5-cak? distribution of the average time (sample mean) between two calls? Find the probability that the average time between two calls is: d) Suppose that a random sample of n -36 calls is observod. According to the CLT, what is the Spts i. Less than 2 hours. ii. More than 3 hoursExplanation / Answer
The average time between two calls is 2 hours so we recieve one call in every two hour interval
a. the pdf for a poisson distribution is as follow:
f(x) = (e^-u) * (u^x) / x! , where u is mean occurrences in an interval and f(x) is probability of x occurrences in an interval, e = 2.718
So probability of 0 calls in a two hour interval = (e^-1)*(1^0) / 0! = (e^-1) = 0.3679
b. We recieve one call every two hours on average, so even if we didnt recieve any call during the last 7 hours, the probability of not recieving any call in the next two hours is still 0.3679 according to the poisson distribution as probability of occurence or non occurence in any interval is independent of the occurrence or non occurrence in any other interval
c. Now the expected time between the 4th and the 5th calls is still 2 hours as probability if occurence is the same for any two intervals of equal length according to the classification of a distribution as a poisson distribution
d. i. Probability of average time < 2 hours means means more than one call in two hours
i.e. P(x>1) = 1- P(x<1) = 1- 0.406 = 0.593, where P(X<1) is the cumulative poisson pdf
ii. Probability of average time > 3 hours means means less than one call in two hours
i.e. P(X<1) = 0.406 where P(X<1) is the cumulative poisson pdf