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. There are other even primes except 2. 3. Multiplicati
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 in Probability section 6.7 in the 7th edition. Both of
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 this week, but not the trickiest aspects.
Functions of Ra
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 of homework must be your own. Homework is generally due
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 homework must be your own. Homework is generally due i
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 homework must be your own. Homework is generally due in
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 normal coins and one trick coin with two heads. You pick
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 from EP4A Section 4.7 page 141
Please hand in: problems
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 have the Bernoulli distribution with p=3/4, = p(1 p) =
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
2k 1 = 4.472, 36 2k 1 = 21.972. 36
and EX 2 =
6 k=1
k2
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 uniform on (0,1) then EX = 1/2, but E (1/X ) =
1 0 1 0 0
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 number of female customers arriving in a given hour. We k
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 (1 p)n ke n n!
=
n=k
=
(p)k [(1 p)]nk e k! (n k )! n= k
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 antiderivative we see that Ex = when 2. 3.5.8 Let X and Y be t
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 equation (3.1) the density function of r(X ) is given by f
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, we have 0.251 P (X = 0) + P (X = 1) = e0.25 + e0.25 = 0
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 0.9 but errors occur with probability 0.1, we have P (A|B