The daily tonnage of garbage handled by the EnviroSafe Landfill Co. is represented as the outcome of a random variable having some triangular distribution, as

This distribution is represented graphically as follows:

Enviro-Safe is in the process of analyzing whether or not they need to expand their facilities. It wants an estimate of the expected daily tonnage of garbage that it handles. It has collected 4 years of daily observations on tonnage handled ( \(n=1,460\) observations) and provides you with the following summary statistic:
\(\sum_{i=1}^{n} x_{i}=29,200\).
You may treat the observations as outcomes of iid random variables.

(a) Use \(\bar{x}=n^{-1} \sum_{i=1}^{n} x_{i}\) to provide an estimate of \(a\), the expected tonnage of garbage handled.

(b) Based on the LLCLT, define an asymptotic distribution for \(\bar{x}\). You should be able to identify a numerical value for the variance of the asymptotic distribution.

(c) Using the asymptotic distribution, how probable is it that the outcome of \(\bar{x}\) will be within .05 tons of the actual expected value of daily garbage tonnage handled?

(d) Use Van Beeck's inequality to provide an upper bound to the approximation error in the probability value that you assigned in part (c). Are you reasonably confident that you provided Enviro-Safe with an accurate "guess" of the expected daily tonnage? (Enviro-Safe management said that they would be satisfied if they could get an estimate that was "within \(\forall \pm 1\) tons of the actual expected daily tonnage.") Explain.

(e) Using your estimate \(\hat{a}=\bar{x}\) to estimate the density function, as \(f(x ; \hat{a})\), what is your estimate of the probability that tonnage handled by Enviro-Safe will exceed 21 tons on any given day?

Transcribed Image Text:

X~ f(x: a) = [(.5-25a) + 25x]Ia-2.a] (X) + [(.5+ 25a) - 25x]I (aa12)(x)