Stats 510
Homework 1
1. (Sets and probabilities) Let the sample space S be the real line. Suppose that a sigma algebra B contains all half-closed intervals of the form. (−∞, a] where a is a rational number. (Note: (−∞, a] = {x|x ≤ a}). Show that the following sets are elements of B
(i) all singleton sets {a} where a is a rational number.
(ii) all singleton sets {a} where a is a real number.
(iii) all intervals of the form. (a,b), [a,b), and [a,b], where a and b are rational numbers.
(iv) all intervals of the form. (a,b), [a,b), and [a,b], where a and b are real numbers.
(v) Give an example of an element of B that is neither empty set, nor S, nor any of the forms mentioned above.
(Hint: This is done by verifying that the set in question can be obtained from known elements of B via countably many set operations. Use the fact that any real number can be constructed as the limit of a sequence of rational numbers).
2. (Optional) Let A1 ⊃ A2 ... ⊃ An ⊃ be a decreasing sequence of subsets in a sigma algebra B associated
with a sample space S. The limit of this sequence of subsets is defined as
Let P be a probability function on B. Use the axioms of probability to show that
(i) If A is empty set, then P(An ) → 0 as n tends to infinity.
(ii) In general, show that P(An ) decreases to a limit that is P(A). We write P(An ) ↓ P(A).
3. (Counting) Do problems 1.20, 1.23, 1.24.
4. (i) Suppose that we had a collection of five numbers, {1, 2, 7, 8, 14}. What is the probability of drawing, with replacement, the unordered sample {2, 7, 7, 8, 14}? (Hint: Look at (ii)).
(ii) Verify that an unordered sample of size k, from m different numbers repeated k1 , k2,..., km times, where k1 + k2 + ... + km = k, has ordered components.
(iii) Use the result of the previous part to establish the identity
5. (Conditional probabilities) Do problems 1.36, 1.37, 1.38.