18.440 Problem Set 9

Due in class Wednesday November 19; late work will not be accepted. You can discuss problems with anyone, but you should write solutions entirely on your own.

1. (20 points) Suppose X is a random variable uniformly distributed over [−1, 1]. Calculate the kth moment

m_{k}(X) = E(X^{k}),

for any integer k ≥ 0. (Hint: even and odd values of k are different.)

2. (20 points) Suppose X and Y are independent random variables, and that all the higher moments m_{p}(X) and m_{q}(Y) are defined. Find a formula for the kth moment m_{k}(X + Y) in terms of the various m_{p}(X) and m_{q}(Y). (Hint: the first example is the formula E(X + Y) = E(X) + E(Y) for the first moments.)

3. (30 points) Suppose X_{1}, X_{2}, X_{3}, … are independent random variables, each uniformly distributed over the interval [−1, 1]. Define

S_{n} = X_{1} + X_{2} + ⋯ + X_{n},

a random variable taking values in [−n, n].

a) Calculate E(S_{n}).

b) Calculate V ar(S_{n}).

c) Calculate the kth moment m_{k}(S_{2}). (Notice that this only asks about S_{2} = X_{1} + X_{2}. You should use the formulas you found in problems 1 and 2. Even and odd k are different. You can use the fact that

∑_{p=1}^{q} \binom{2q}{2p − 1} = 2^{2q−1}.)

4. (10 points) For the random variables in Problem 3, calculate probability density functions f_{S_{k}} for k = 1, 2, and 3. Graph them carefully enough to show that f_{S_{2}} has a derivative everywhere.

5. (20 points) Suppose f_{X} is the probability density function for a random variable uniformly distributed over [−1, 1], and f is any continuous function. The point of this problem is to show that the convolution

(f ∗ f_{X})(x) = ∫_{−∞}^{∞} f(t)f_{X}(x − t) dt

has a derivative.

a) Show that

(f ∗ f_{X})(x) = \frac{1}{2} ∫_{x−1}^{x+1} f(t) dt.

b) Show that

(f ∗ f_{X})(x + Δx) − (f ∗ f_{X})(x) = \frac{1}{2} ∫_{x+1}^{x+1+Δx} f(t) dt − \frac{1}{2} ∫_{x−1}^{x−1+Δx} f(t) dt.

(This formula is easy to interpret as long as Δx > 0. You can just look at that case if you prefer. For negative values of Δx, you need to use conventions from calculus like ∫_{a}^{b} = − ∫_{b}^{a}; then the formula is still true.)

c) Show that

\frac{d}{dx}[(f ∗ f_{X})](x) = \frac{1}{2}(f(x + 1) − f(x − 1)).

d) Suppose g is a function on ℝ having a continuous derivative and vanishing outside some finite interval. Explain why it ought to be true that

\frac{d}{dx}[f ∗ g] = f ∗ \frac{dg}{dx}.

(This formula really is exactly true, but I’m only asking for a plausibility argument.) Can you connect this formula to the one in part (c)?