Probability Homework 3
Jiawei Sun
Exercise 1
(i) We know that with p(x) is the density, E(X) = R xp(x)dx. Lets
calculate the following integral, let (x )/( 2) = t, then
R
(x2 )
1
xe 22 dx =
2
1
1
=
=
2
( 2t + )et dt
R
2
et dt
R
(ii)
E(X) =
xex dx
0
= xe
Probability, homework 2, due September 24.
Exercise 1. Suppose that is an innite set (countable or not), and let A be the
family of all subsets which are either nite or have nite complement. Prove that
A is not a -algebra.
Exercise 2. Let A be a -algebra,
Probability, homework 3, due October 1.
Exercise 1. Calculate E(X) for the following probability measures PX .
(i)
(ii)
(iii)
(iv)
2
2
1
PX has Gaussian density 2 e(x) /(2 ) , for some > 0 and R;
PX has exponential dentity ex 1x0 for some > 0;
PX = pa + q
Probability, homework 4, due October 15.
Exercise 1. Let X be uniformly distributed on [0, 1] and > 0. Show that
1 log X has the same distribution as an exponential random variable with parameter .
Exercise 2. Let X1 , . . . , Xn be bounded, independent a
Probability, homework 5, due October 22.
Exercise 1. Let X be uniform on (, ) and Y = sin(X). Show that the density
of Y is
1
1[1,1] (y).
1 y2
Exercise 2. Let (X, Y ) be uniform on the unit ball, i.e. it has density
f(X,Y ) (x, y) =
Find the density of
1
Probability, final training.
Eight exercises of this type will be proposed, you will have 100 minutes. Grade
will be scaled so that six exercises perfectly solved will give maximal score 100/100.
Exercise 1. Let X be a real random variable uniform on [0,
Probability, homework 5, due October 14.
Exercise 1. Calculate E(X) for the following probability measures PX .
(i) PX = pa + qb where p + q = 1, p, q 0 and a, b R;
n
(ii) PX is the Poisson distribution: PX (cfw_n) = e n! for any integer n 0, for
some > 0
Probability, homework 4, due October 7.
Exercise 1. Suppose that 3 percent of men and .5 percent of women are color
blind. A color-blind person is chosen at random. What is the probability of this
person being male? Assume that there are an equal number o
Probability, homework 9, due December 9.
Exercise 1. Let X be a real random variable, with Poisson distribution with
parameter . Calculate the characteristic function of X . Conclude that (X )/
converges in distribution to a standard Gaussian, as .
Exer
Probability, homework 6, due November 4.
Exercise 1. Let X be a standard Cauchy random variable. What is the density
of 1/X?
Exercise 2. Let X1 , . . . , Xn be bounded, independent and identically distributed
random variables such that E(X1 ) = 0, E(X12 )
Probability, homework 7, due November 11.
Exercise 1. Let X be uniform on (, ) and Y = sin(X). Show that the density
of Y is
1
p
1[1,1] (y).
1 y2
Exercise 2. Let (X, Y ) be uniform on the unit ball, i.e. it has density
1
if x2 + y 2 1,
f(X,Y ) (x, y) =
Probability, homework 1, due September 16.
Exercise 1. Prove whether the following sets are countable or not.
(i) All intervals in R with rational endpoints.
(ii) All circles in the plane with rational radii and centers on the diagonal x = y.
(iii) All se
Basic Probability Assignment 1
Provide complete justication. You may use results proved in the two texts.
Due date March 4.
Question 1: The event A is said to be repelled by the event B if P (A|B) <
P (A). and to be attracted by B if P (A|B) > P (A). Show
Basic Probability Assignment 2
Provide complete justication. You may use results proved in the two texts.
Due date April 15.
Question 1: Suppose that X1 , X2 , . . . Xn are independent identically distributed
continuous outcome random variables which have
Probability, midterm training.
Eight exercises of this type will be proposed, you will have 100 minutes. Grade
will be scaled so that six exercises perfectly solved will give maximal score 100/100.
Exercise 1. Let (, A, P) be a probability space. Prove th
Probability, homework 3, due September 30.
Exercise 1. Let A be a -algebra, P a probability measure and (An )n1 (resp.
(Bn )n1 ) be a sequence of events in A which converges to A (resp. B). Assume
that P(B) > 0 and P(Bn ) > 0 for all n. Show that
(i) limn
Probability, homework 8, due December 3.
Exercise 1. Let X be a random variable with density fX (x) = (1 |x|)1(1,1) (x).
Show that its characteristic function is
2(1 cos u)
X (u) =
.
u2
Exercise 2. Let f be a continuous fuction on R, and assume that (Xn )
Probability, homework 6, due October 29.
Exercise 1. Let (Xn )n1 be a sequence of Gaussian random variables, Xn having
2
2
mean n and variance n . Assume n R and n 2 R. Prove that Xn
converges in distribution to a Gaussian random variable with mean and va
Probability, homework 7, due November 19.
Exercise 1. Let (, A, P) be such that is cointable and A = 2 . Prove that
almost sure convergene and convergence in probability are the same on this probability space.
Exercise 2. Let (Xi )i1 be i.i.d. Gaussian wi
Probability, homework 8, due November 26.
Exercise 1. Let Y be a positive and integrable random variable on (, A, P) and
G a sub -algebra of A. Show that |E (Y | G)| E (|Y | | G).
Exercise 2. Let Y be a positive and integrable random variable on (, A, P)
Probability, homework 9, due December 10.
Exercise 1. Let (Xk )k0 be i.i.d. random variables, Fm = (X1 , . . . , Xm ) and
m
Ym = k=1 Xk . Under which conditions is Y a F-submartingale, supermartingale,
martingale?
Exercise 2. Let (Sn )n0 be a (Fn )-martin
Probability Homework 1
Jiawei Sun
Exercise1
(i) We have total 5! choices, and 3! 2! choices of which are what we want.
3! 2!
So the probability is
5!
2
(ii) Pick 2 from 11, we have C11 choices. Pick 4 from the remaining 9, we
4 choices, and there are 5 le
probability homework 2
Jiawei Sun
Exercise 1
1 If is an innite countable set, then Z, so we just need to consider
=
the case when = Z: Let Ai = cfw_i. Obviously, for any i, Ai A, but
Ai = cfw_1, 2, 3, . A. So A is not a -algebra.
i=1
2 If is uncountable,
Probability Homework 4
Jiawei Sun
Exercise 1
X is uniformly distributed on [0, 1], so FX (x) = x. Also, for > 0,
1
1
log X < x X > ex . Let log X = Y , Since 0 X 1,
Y 0. Thus on [0, ) we have
1
P(Y < x) = P( log X < x) = P(X > ex )
= 1 P(X ex ) = 1 ex
=
Probability Homework 5
Jiawei Sun
Exercise 1
Since sin x in (, ) is not monotone, we should split it. Clearly,
1 < Y < 1. sin x is monotone on (, /2), (/2, /2) and
(/2, ).
(1) When < x < , then 0 < + x < , and in this case
2
2
1
1 < y < 0. Since FX (x) =
Probability 6
Jiawei Sun
Exercise 1
We know that2 the characteristic function of Gaussian distribution is
t2
f (t) = eit 2 . Also, we know n , n , that means
, > 0, N , s.t. for n N , |n | , |n | .
Also, we know |n + | must be bounded since n (i.e. M ,
Probability Homework 7
Jiawei Sun
Exercise 1
Since almost surely convergence always implies convergence in probability, we only show that in this particular space, convergence in
probability also implies almost surely convergence.
We know that is countabl
Probability Homework 8
Jiawei Sun
Exercise 1
Note that (x) = |x| is a convex function. Then applying to Jesens
inequality, we have
|E(Y |G)| E(|Y |G)
Exercise 2
According to the denition of conditional expectation, for any Z
L2 (, H, P), we have
E(Y Z) =
Probability Homework 9
Jiawei Sun
Exercise 1
We must have Xk L1 , and we suppose E(Xi ) = . Then we have for
any n
E(Yn |Fn1 ) = E(Yn1 Xn |Fn1 ) = Yn1 E(Xn |Fn1 ) = Yn1
Clearly, when 1, Yn is a sub-martingale; when 1, Yn is a
super-martingale, when = 1,
Probability, homework 2, due September 23.
Exercise 1. Suppose that is an infinite set (countable or not), and let A be the
family of all subsets which are either finite or have finite complement. Prove that
A is not a -algebra.
Exercise 2. Let (An )n0 be