The plots of co-variance functions should start from 0-shift

Question

The following was my question given by my teacher,

1. Generate a sequence of N = 1000 independent observations of random variable with distribution: (c) Exponential with parameter λ = 1 , by inversion method.
2. Present graphically obtained sequences(except for those generated in point e) i.e. e.g. (a) i. plot in the coordinates (no. obs., value of the obs) ii. plot in the coordinates (obs no n, obs. no n + i) for i = 1, 2, 3. iii. plot so called covariance function for some values. i.e. and averages:

I have written the following code,

(*****************************************************************)
(*Task 01(c) and 02(a)*)
(*****************************************************************)
n = 1000;

taskC = Table[-Log[RandomReal[]], {n}];

ListPlot[taskC, AxesLabel->{"No. obs", "value of the obs"}]

i = 1;
ListPlot[Table[
    {taskC[[k]], taskC[[k+i]]}, 
    {k, 1, n-i,1}],
AxesLabel->{"obs.no.n", "obs.no.n+1"}]

i++;
ListPlot[Table[
    {taskC[[k]], taskC[[k+i]]},
    {k, 1, n-i,1}],
AxesLabel-> {"obs.no.n", "obs.no.n+2"}]

i++;
ListPlot[Table[
    {taskC[[k]], taskC[[k+i]]},
    {k,1,n-i,1}],
AxesLabel->{"obs.no.n", "obs.no.n+3"}]

avg = (1/n)*Sum[taskC[[i]], {i,n}];

ListPlot[Table[1/(n-tau) * Sum[(taskC[[i]]-avg)*(taskC[[i+tau]] - avg), n], {tau, 1,100}], 
    Joined->True, 
    AxesLabel->"Covariance Function"]

He has commented,

The plots of co-variance functions should start from 0-shift. Note that for larger than 0 shifts you are estimating co-variance between independent observations which is zero, while for 0 shift you are estimating variance of observation which is large. Thus the contrast between these two cases is a clear indication that the observations are uncorrelated.

What did I do wrong?

How can I correct my code?

Answer 1

Zero-shift means calculating the covariance for tau = 0, which is simply the variance.

Labeled[ListPlot[Table[{tau,
    1/(n - tau)*Sum[(taskC[[i]] - avg)*(taskC[[i + tau]] - avg), {i, n - tau}]},
   {tau, 0, 5}], Filling -> Axis, FillingStyle -> Thick, PlotRange -> All,
  Frame -> True, PlotRangePadding -> 0.2, AspectRatio -> 1],
 {"Covariance Function K(n)", "n"}, {{Top, Left}, Bottom}]

Variance[taskC]

0.93484

Covariance[taskC, taskC]

0.93484

(* n = 1 *)
Covariance[Most[taskC], Rest[taskC]]

0.00926913