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I have a fairly involved homework question, I was wondering if I could get some help.

There are two types of phone calls arriving at a switch, long-duration and short-duration. Each day the number of calls is observed up to and including the first long-duration call. This was done for 8 days and the information gathered is as follows: 15, 6, 9, 12, 18, 3, 15, 21.

a) Compute the probability of observing long and short duration calls.

b) If 100 calls are made on each day, compute the mean and the variance of the number of long and short duration calls on each day.

c) If the average length of a long duration call is 60 minutes and the average length of a short duration call is 3 minutes, compute the effective average length of a call.

The way I did part a was by adding 1/15 + 1/6 ... (since this is the prob of getting a long-duration with the sample data provided) and then dividing by 8 to get 0.116 prob of observing a long-duration. and 1-0.116 to observe a short-duration.

To do b I took (1/15)x100, (1/6)x100, ... to find the number of long-duration calls in 100 and then found the mean and variance of those numbers. I'm not confident about this method. I got a mean of 11.636 and variance of 91.553 for long-duration (the variance should be the same of short and long duration, yeah?) But now that I think about it looks like the question is asking about each day separately.

I think c was pretty simple. I just took (1/15)x60 + (14/15)x30 + (1/6)x60 + (5/6)x30 +... and then divided that sum by 8 to get 6.591 as the average length of a call.

I know there is a lot there, any help is much appreciated. Thank you!

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