Honors Introduction to Analysis I Homework II
Solution February 9, 2009
Problem 1 If A, X , Y are sets then AX denotes the set of all functions from X to A and X Y denotes the Cartesian product of X a
Honors Introduction to Analysis I Homework I
Solution February 2, 2009
Problem 1 1. There is a positive integer that either doesnt have a prime factorization, or it has several prime factorizations. 2
Math 4710: Homework 10 Due Friday, December 5, 2008
When possible, solve these problems using the theorems we've studied from Grinstead and Snell instead of the more ad hoc methods of EP4A.
Problems f
Math 4710: Homework 6 Due Friday, October 17, 2008
The primary references for this weeks assignment are Prof. Durretts Essentials of Probability sections 3.6 and 3.7 as well as Prof. Ross First Course
Math 4710: Homework 5 Due Friday, October 3, 2008
With our exam Monday October 6, well be posting solutions to these problems Friday morning after class. The exam will include the material we study th
Math 4710: Homework 4 Due Friday, September 26, 2008
Please remember that expressing your reasoning or proof clearly is very important! You may discuss problems with other students but your writeups o
Math 471: Homework 2 Due Friday, September 12, 2008
Please remember that expressing your reasoning or proof clearly is very important! You may discuss problems with other students but your writeups of
Math 471: Homework 1 Due Friday, September 5, 2008
Please remember that expressing your reasoning or proof clearly is very important! You may discuss problems with other students but your writeups of
Prelim I, 6 hours
SHOW ALL WORK! (1)[10 pts] Events A, B, C are independent. (a) Show that Ac , B c , C c are independent. (b) Show P (A B C ) = 1 (1 P (A)(1 P (B )(1 P (C ).
(2)[10 pts] You have 9 no
Prelim II, 50 minutes
SHOW ALL WORK! (1) [30] Calculate the expectation and variance of a Gamma (7, ) random variable. Hint: you can do this any way you want, but one way is to use its relationship wi
FINAL EXAM, Math 471
(1) [11 pts] Give an example of a sequence of random variables cfw_Xn that converges in probability, but not with probability one. Be explicit about what the event space S is and
Prelim II, 50 minutes
SHOW ALL WORK! (1) [30 pts] Consider the Normal(0,1) random variable X . Find the density of the random variable Y = X 4. d d P (X 4 < y ) = P (y 1/4 < X < y 1/4 ) dy dy
y 2 d 1
Prelim II 2005 Solutions
(1) For the mean, see Example 1.7 in Section 4.1 or use the fact that a gamma(n, ) is a sum of n independent exponential() random variables:
7
E (gamma(7, ) =
i=1
7 1 =.
For
MATH 471 HW 13 Solutions
Pengsheng Ji
December 7, 2006
5.2.4 (a) Let Xi = 1 if she answers question i correctly, =0 otherwise. S48 = X1 + + X48 is the number of questions she answers correctly. The Xi
MATH 471 HW 12 Solutions
Pengsheng Ji
December 7, 2006
5.1.2 Note EX = 4 1/2 = 2, var(X ) = 4 (1/2)2 = 1. Chebyshevs upper bound is P (|X 2| 2) var(X ) 1 =. 2 2 4
The exact probability is P (|X 2| 2)
MATH 471 HW 11 Solutions
Pengsheng Ji
November 28, 2006
4.4.3 Let X be the highest number that was rolled. We nd P (X = 1) = 1/36, P (X = 2) = 3/36, . . ., P (X = k ) = (2k 1)/36. Thus,
6
EX =
k=1
k
2
MATH 471 HW 10 Solutions
Pengsheng Ji
November 7, 2006
4.2.13 Observing tx
x2 2
= (x t)2 + t2 /2, we have EetX = = et
2
1 2 etx ex /2 dx 2 1 2 e(xt) /2 dx 2 = et
2 /2
.
4.2.20 EXY = 4.2.25 If X is un
MATH 471 HW 9 Solutions
Pengsheng Ji
November 7, 2006
1 (a) Let N be the number of customers arriving in the store during a typical hour. It has a Poisson distribution with parameter . Let K be the nu
MATH 471 HW 8 Solutions
Pengsheng Ji
November 7, 2006
3.8.10 Let N be the number of successes. Then P (N = k |X = n) = Cn,k pk (1 p)nk for k n, so
P (N = k ) =
n=k
P (N = k |X = n)P (X = n) Cn,k pk (
MATH 471 HW 7 Solutions
Pengsheng Ji October 31, 2006
4.1.16 By denition, we have EX =
1
x ( 1)x dx =
1 2 1 x |1 = 2 2
Note that the last calculation equality is valid only for > 2. From the antider
MATH 471 HW 6 Solutions
Pengsheng Ji
TA Oce Hours: 4:05-5:05 PM Tuesday 5:20-6:20 PM Thursday 218 Mallot Hall
3.3.5 X has density function f (x) = ex and r(x) = x1/ has the inverse s(y ) = y . By equa
MATH 471 HW 5 Solutions
Pengsheng Ji
3.1.11 We know Poisson distribution can be used to approximate the binomial distribution. In this problem, p=0.01, n=25, = np = 0.25. Using Poisson approximation,
MATH 471 HW 4 Solutions
Pengsheng Ji
Oce Hours: 4:05-5:05 PM Tuesday 5:20-6:20 PM Thursday 218 Mallot Hall
2.4.6 Let A=1 was received, B=1 was sent. Since they are received as sent with probability 0.
4130 HOMEWORK 8 Due Tuesday May 3
(1) Let fn : A R be functions which converge uniformly on A to a function f . Let x0 be a cluster point of A. Suppose limxx0 fn (x) exists for all n. Let Ln = limxx0
4130 HOMEWORK 7 Due Tuesday April 13 (1) Let D R. Let f, g : D R and let a be a cluster point of D. Suppose limxa f (x) = L and limxa g (x) = M . Show that limxa f (x)g (x) = LM . First, we show that
4130 HOMEWORK 6 Due Thursday April 1
(1) Let A R. A point x A is called isolated if it is not a cluster point of A. (a) Can an open set have an isolated point? Can a closed set have one? An open set U
MATH 4130 FINAL EXAM Math 4130 nal exam, 18 May 2010. The exam starts at 7:00 pm and you have 150 minutes. No textbooks or calculators may be used during the exam. This exam is printed on both sides o