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
=
=
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
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 h
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 tha
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 .
Ex
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 dis
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
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 )/
conv
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
per
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,
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 rationa
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.
Ex
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 rationa
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
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 dis
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 distributi
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.
Exer
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.
Ex
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 dist
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 probabili
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
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
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
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 <
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 < +
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 | .
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 impl
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 conditiona
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-m
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. Pro