You walk to Stevens Way and wait for a bus. Instead of getting on the bus, you r

Question

You walk to Stevens Way and wait for a bus. Instead of getting on the bus, you record your waiting time Ti. You repeat this for a total of 50 buses, obtaining T2... Tso. Assume they arrive as a Poisson process with buses per minute. (a) What is the "population" here? (b) According to your assumption, what distribution should the wait times follow? (c) Find an unbiased estimator for the mean of this distribution using only the data from the first 20 buses. (d) Find another estimator using all the data. (e) Before doing any further computations, consider your estimators from (c) & (d). Guess which one is "better" and why. (Your score on this problem part depends on your explanation, reasoning & justification.) (f) Calculate the MSE of each. Try to reduce the problem to expectations or variances involving a single datum (g) What is the purpose of these Part II tasks & questions? Why did we have you do this?

Explanation / Answer

a) population = total no. of buses in a day

b) it should follow a poisson's distribution as buses' arrival rate follows poisson's distribution.

c) from 20 buses we can find out lambda= arrival rate= no. of buses arriving per minute

d) from all the data we can find out the probability that arrival of 50 buses will follow a rate same by 20 buses.

e) the estimator calculated in c or lambda is a better estimator because it's an unbiased estimator. the probability calculated in part d will keep on changing as we change the no. of buses.

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