Warmup Problems 3 (18.440)
1. Show, by direct "brute force" computation, that E[N 2 ] =
P∞
i=1
P{N 2 ≥ i}.
Hint: To do this directly, interchange the following summation. (Why is N 2 ≥ i if and only if N ≥
∞
X
i=1
P{N 2 ≥ i} =
∞
X
P{N ≥
√
i} =
∞
∞
X
X
√
i?)
P{N = k}
√
i=1 k=d ie
i=1
√
As when exchanging limits of integration, it may help to draw a plot like the one below of k = d ie. We
are taking the sum over the integer-valued points (i, k) in the shaded region that goes up to ∞. What is the
range of i at each k?
P∞
What is an easier way of P
showing that E[N 2 ] = i=1 P{N 2 ≥ i} if you are allowed to use the result in
∞
problem 5 that says E[X] = i=1 P{X ≥ i}?
Also, can you prove pset problem 5 by applying the result of the warmup problem? Why or why not?
Hint: Use the fact that N and X take only integer values to decide if this is possible.
Some things to watch out for
1a. Pairwise independent vs. other forms of "independent" (pairwise is weakest).
e.g., let X, Y be two coin flips ("0" if tails, "1" if heads), and let Z = X + Y mod2. Then (X, Y ), (Y, Z)
and (X, Z) are all distributed like two independent coin flips. However, (X, Y, Z) are obviously not jointly
independent, since knowing any two of the numbers will tell you the third number.
Note: In problem 1 of the homework, you need to show both (1) that the R.V.’s are pairwise independent
and also that they are (2) NOT jointly independent.
1b. Pairwise disjoint vs. other forms of "disjoint" (pairwise is strongest).
e.g., we may have ABC = ∅, but none of AB, BC, and AC are empty:
1