Homework 1

Probability, Fall 2013
Homework #1
Due: Thursday, October 17th, 2013 in class
Fun Probabilistic Questions
1. Recall that a geometric random variable is a discrete one taking values in the set
{1, 2, 3, . . . , }, with P (X = k) = p(1 − p) k−1 where 0 ≤ p ≤ 1.
(i) The random number X should always be thought of as the number of independent
trials it takes before an experiment succeeds, where p is the probability of success
on any given trial. Give a brief explanation of why this is from the formula for
P (X = k).
(ii) Compute the mean of X.
(iii) Compute the variance of X.
2. A bowl contains twenty cherries, exactly fifteen of which have had their stones removed.
A greedy pig eats five whole cherries, picked at random, without remarking on the
presence or absence of stones. Subsequently, a cherry is picked at random from the
fifteen.
(i) What is the probability that the cherry contains a stone?
(ii) Given that this cherry contains a stone, what is the probability that the pig
consumed at least one stone?
3. If m students all born in 1991 are attending a lecture, show that the probability that
at least two of them share a birthday is p = 1 − (365)!/((365 − m)!365 m ). Convince
yourself that p > 1/2 when m = 23 and p < 1/2 when m = 22.
4. A pack contains m cards labelled 1, 2, . . . , m. The cards are dealt out in a random
order, one by one. Given that the label of the kth card dealt is the largest of the first
k cards dealt, what is the probability that it is also the largest of the pack?
5. Let’s pretend that you love collecting the toys from Happy Meals (you don’t have to
pretend if you don’t want to). Suppose that at present there are K different types
of Happy Meal toys available. When you buy a Happy Meal, each particular toy has
probability 1/K of being in the little box. You buy one Happy Meal each day.
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 .

If there exists a function f(x) such that
F (x) =
∫ x
then f is called the density function of F .
−∞
f(t) dt,
9. For each of the following, indicate if it is a distribution function and if so give the
corresponding density function.
(i)
(ii)
F (x) =
F (x) =
(iii) F (x) = e x /(e x + e −x ) for x ∈ R
(iv) F (x) = e −x2 + e x /(e x + e −x ) for x ∈ R
{
1 − e
−x 2 , x ≥ 0
0, x < 0
{
e −1/x , x > 0
0, x ≤ 0
10. Let X have distribution function
⎧
⎨ 0, x < 0
P (X ≤ x) = x/2, 0 ≤ x ≤ 2
⎩
1, x ≥ 2
and let Y = X 2 . Find
(i) P (1/2 ≤ X ≤ 3/2)
(ii) P (1 ≤ X < 2)
(iii) P (Y ≤ X)
(iv) P (X ≤ 2Y )
(v) P (X + Y ≤ 3/4)
(vi) the distribution function of Z = √ X.
11. Suppose X has a continuous density f, with P(α ≤ X ≤ β) = 1 for some −∞ ≤ α <
β ≤ ∞, and g is a function that is strictly increasing and differentiable on (α, β). Show
that g(X) has the density function
f(g −1 (y))
g ′ (g −1 (y)) ,
y ∈ (g(α), g(β)).

12. (i) Suppose X has a normal distribution with mean µ and variance σ 2 . Use the last
exercise to compute the density of exp(X). (The answer is called the lognormal
distribution, and is important in finance).
(ii) Suppose X has a normal distribution with mean 0 and variance 1. Use the last
exercise to compute the density of X 2 . (The answer is called the chi-squared
distribution.)
13. Show that if F (x) = P(X ≤ x) is continuous then Y = F (X) has a uniform distribution
on (0, 1), that is P(Y ≤ y) = y for y ∈ (0, 1).
This gives you an extremely useful method of simulating random variables on the
computer. If you can compute F −1 , then you can ask the computer to generate a
uniform random variable Y for you (all computers have a way of doing this, but the
techniques behind it could take up an entire class), and then X = F −1 (Y ) will have F
as its distribution function.
14. If a distribution function is not continuous, it’s because there is at least one number
x 0 (there may be more) such that
lim F (x) =: F (x 0 +) > F (x 0 −) := lim F (x).
x↓x 0 x↑x0
Such numbers x 0 are called atoms of the distribution because there is a positive
probability of the number being picked. In fact
Convince yourself that
P(X = x 0 ) = F (x 0 +) − F (x 0 −).
(i) a distribution function with discontinuities does not have a density function,
(ii) a distribution function can have at most countably many discontinuities.
There’s no need to write anything for this problem, it’s for your own edification. Just
make sure you think about it for a little bit.
15. A distribution function can be continuous but still not have a density function. Such
distributions are called singularly continuous, and are kind of wicked. Here’s a
simple and classical example of how to construct one called the Cantor staircase:
(i) Start by constructing the Cantor middle-thirds set. Let C 0 = [0, 1], and to
construct C 1 remove the “middle-third interval” (1/3, 2/3) of C 0 . This leaves
C 1 = [0, 1/3] ∪ [2/3, 1]. In general construct C n+1 from C n by removing the
middle-third of each interval in C n . Draw a picture of C n for n = 0, 1, 2, 3. You
see that the set gets sparser and sparser very quickly. In the limit though there
is a non-empty set remaining, let’s call it C ∞ that is referred to as the Cantor
middle-thirds set. Look it up on Wikipedia if you haven’t seen it before.

(ii) Now let f 0 , f 1 , f 2 , . . . be the uniform densities on C 0 , C 1 , C 2 , . . .. That is
{
Kn , x ∈ C
f n (x) =
n
0, x ∉ C n
where K n is some constant that doesn’t depend on x. What does K n have to be?
(Hint: recall that the area under a density function must always be one).
(iii) Now let F 0 , F 1 , F 2 , . . . be the distribution functions corresponding to f 0 , f 1 , f 2 , . . ..
Give a quick sketch of F 0 , F 1 , F 2 , and F 3 .
As n → ∞ the distribution functions F n get steeper and steeper on smaller and
smaller sets. In the limit what you get is a distribution function F ∞ supported
that is “infinitely steep” on a set of “Lebesgue measure zero”. Because the
distribution function F ∞ is supported on the set C ∞ that has measure zero,
it can’t possibly have a density function.
This illustrates another important concept. Even though F ∞ is the limit of the
distribution functions F n , which all have densities, F ∞ itself does not have a
density.