Homework 1

(i) Find the mean number of days that elapse between the acquisition of the jth new
toy and the (j + 1)st new toy.
(ii) Find the mean number of days which elapse before you have a the full collection
of toys.
σ-Algebras and Random Variables
6. Let Ω = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T }, let XY Z denote a
generic element of Ω, and define the following random variables on Ω:
A(XY Z) = 1, C 1 (XY Z) = 1 {X = H} , C 2 (XY Z) = 1 {Y = H} , C 3 (XY Z) = 1 {Z = H} .
(i) What is σ(A)?
(ii) What is σ(C 1 , C 2 )?
(iii) What is σ(C 1 , C 2 , C 3 )?
(iv) Is C 2 measurable with respect to σ(C 1 − C 2 + 2C 3 )?
Probability Measures
7. Let A and B be events with P(A) = 3/4 and P(B) = 1/3. Show that 1/12 ≤ P(A∩B) ≤
1/3. Give examples to show that both bounds can be achieved.
8. Let A n , n ≥ 1, be events such that P(A n ) = 1 for all n. Show that P(∩ ∞ n=1A n ) = 1.
(HINT: Use the inequality P(∪A n ) ≤ ∑ P(A n ). This is sometimes called the “union
bound”.)
Distribution Functions
Recall that any function F : R → [0, 1] is a distribution function if it is non-decreasing
and
lim F (x) = 1, lim F (x) = 0.
x→∞ x→−∞
For any distribution function F , there is always a random variable X with
F (x) = P(X ≤ x).
In fact you will see in Question 10 how to actually create, on the computer, a random
variable X with a given distribution function F .