🚀 go-pugleaf

Here are some ideas off the top of my head, that should get you going in
the right direction.  You will need some decent calculus skills, but
that's about it. 

Case 1.  X is a real-valued continuous random variable with density f(x)
such that xf(x) -> 0 as x -> +infinity and as x -> -infinity.  Write
E[|X-c|] as the sum of two integrals, one going from -infinity to c and
the other going from c to +infinity, where neither integral involves
abs. val's anymore.  Differentiate this sum with respect to c and
explore...

Case 2.  x_1, x_2, ... x_n are fixed real numbers (i.e. you have a
sample of size n).  Then c=the sample median minimizes the sum of
absolute deviations |x_1-c| + |x_2-c| + ... + |x_n-c|.  You can do the
cases n=2, 3, and 4 "by hand" by graphing the sum of abs. deviations as
a function of c---it is piecewise linear with bends at the data points
x_1, x_2, ...  What do you have to do to give a rigorous proof for n
data points now? 

Case 3.  X is a discrete random variable taking on the values x_1, x_2,
.... with probabilities p_1, p_2, .... The ideas you explored above
should be useful in proving that the median minimizes E[|X-c|] for this
case.

-BJ