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What expectation can we infer about the daily total collections of the leaf litter from the two patches?

Mathematics Asked by Phi beta kappa on January 5, 2021

Two forest patches have, respectively, $100$ and $200$ teak trees of the same age. In a given season, all trees shed some of their leaves at random. The daily total collections of the leaf litter from the two patches are expected to have

$(1)$ nearly equal means, standard deviations and coefficients of variation

$(2)$ different means, nearly equal standard deviations and coefficients of variation

$(3)$ different means, nearly equal standard deviation and different coefficients of variation

$(4)$ different means, and standard deviations but nearly equal coefficients of variation

What conclusion can we make from the above data? What I know is that the mean $mu = dfrac {sum x} {n}, $ standard deviation $sigma = sqrt {dfrac {sum x^2} {n} – left (dfrac {sum x} {n} right )^2}$ and coefficients of variation $nu = dfrac {sigma} {mu},$ where $n =$ sample size (in this case the number of trees in each of the forest patch) and $sum x$ represents the total number of leaves collected from each forest patch in a day. Now how do I proceed? Any help in this regard will be appreciated.

Thanks for your time.

One Answer

The point of the problem is that the number of leaves shed by each tree on a given day is to be considered a random variable, and all the variables are independent and identically distributed. So in one case, we have the sum of $100$ i.i.d. random variables, and in the other case we have the sum of $200$ i.i.d. random variables. Let the mean and variance of each be $mu$ and $sigma^2$ respectively.

By linearity of expectation, the mean of the first patch is $100mu$ and that of the second patch is $200mu$. Also, the variance of the sum of independent random variables is the sum of the variances, so the variance of the first patch is $100sigma^2$ and the standard deviation is $10sigma$; the standard deviation of the second patch is $10sigmasqrt2$.

Correct answer by saulspatz on January 5, 2021

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