Math 650. Homework 2. Due 9/5/02

1.[3, p. 11] A gambler has an initial stake of one dollar. Calculate the probability of ruin at times 1, 3, and 5. Show that the chance of eventual ruin is at least 70%.

2.[4, p. 12] Show that the Rademacher functions satisfy

∫ 1 0  Ri1 Ri2… Rin  dx = 0 or 1

for any sequence i₁≤ i₂ ≤ … ≤ i_n. When is the answer 1?

3.[11, p. 13] Let C be the set of all numbers w on the interval [0,1] which can be written in the form

w = ∞ ∑ k=1  ak 3k

with a_k=0 or 2. This set is called the Cantor set. Show that C can be constructed by the following procedure: from [0,1] remove the middle third (1/3,2/3); from the remainder (that is, the intervals [0,1/3] and [2/3,1]) remove the middle thirds, and so on, ad infinitum. What remains is the Cantor set.

4.[12, p. 13] Show that the Cantor set is of Lebesgue measure zero.

5.[20, p. 14] Let f be a continuous, non-negative, monotone function on the unit interval I=[0,1]. Prove Chebyshev's inequality

mL({w ∈ I | f(w) > a}) < 1 a ∫ 1 0  f(x) dx