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.
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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