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 $100\mu$ and that of the second patch is $200\mu$. Also, the variance of the sum of independent random variables is the sum of the variances, so the variance of the first patch is $100\sigma^2$ and the standard deviation is $10\sigma$; the standard deviation of the second patch is $10\sigma\sqrt2$.