MATH 230/STAT 230: Homework 9
Due on 11/24, in class
Read Section 4.5-4.6, and Section 5.1
Problem 1. Suppose an electric circuit has two parts (as shown in the gure). Part I
consists of two branches, whereas Part II has only one branch. The electric circ
MATH 230/STAT 230: Practice Midterm I
Problem 1. A biometric security device using ngerprints erroneously refuses to admit 1 in 100
authorized persons from a facility containing classied information. The device will erroneously
admit 1 in 1,000 unauthoriz
Math 230 / Sta 230
Spring 2015
Second Midterm Exam
April 7, 2015
Time allowed: 75 minutes
Print Name:
Instructions:
(a) This is a closed book exam. You may bring a two-sided A4 paper of your own notes.
(b) Use a calculator when necessary. No cellphones.
(
Math 230 / Sta 230
Spring 2015
Final Exam
May 1, 2015
Time allowed: 180 minutes
Print Name:
Instructions:
(a) This is a closed book exam. You may bring two two-sided A4 paper of your own
notes.
(b) Use a calculator when necessary. No cellphones.
(c) Show
Math 230 / Sta 230
Spring 2015
First Midterm Exam
February 17, 2015
Time allowed: 75 minutes
Print Name:
Instructions:
(a) This is a closed book exam. You may bring a two-sided A4 paper of your own notes.
(b) Use a calculator when necessary. No cellphones
Homework 7 Solution
4.2 4.(a) Let X be the lifetime of the component. Then X Exp() with = 1/10.
(a)
P (X 20) = e20 = 0.1353.
(b)
Median(X) =
log 2
= 6.931.
(c)
1
= 10.
(e) Suppose the two lifetimes are X1 and X2 . Then the probability we want is
SD(X) =
P
Homework 4 Solution
2.4 2. (a) n = 500, p = 0.02, = np = 10. By Poisson approximation
1
P (1 success) = e = 10 e10 = 0.0004540.
1!
(b)
P (2 or fewer successes)
= P (0 success) + P (1 success) + P (2 successes)
0 1 2
e + e + e
0!
1!
2!
2
e
= 1+
2
=
= 0.
Homework 10 Solution
5.3 13. (a) We change the variables from (X, Y ) to the polar coordinate (R, ). We
need to multiply the the Jacobian of r. The joint density of (R, ) is
fR, (r, ) = fX,Y (x, y) r =
r r2
r x2 +y2
e 2 =
e 2.
2
2
for r (0, +) and (0, 2).
Key to Final 2009
Note: Stared questions (*) are not in the range of our nal.
1. (a)
P (A B) = P (A) + P (B) P (A B) = P (A) + P (B) P (A)P (B) =
8
2 2 2 2
+ = .
3 3 3 3
9
(b) We have that
1
P (A B) = P (B|A)P (A) = P (A)
2
= P (A) = 2P (A B)
1
P (A B) =
Key to Final 2013
Note: Stared questions (*) are not in the range of our nal.
1. (a)
P (T > t) = P (T > t|Roof)P (Roof) + P (T > t|Tree)P (Tree)
10
1
1
= et/10 +
2 10 + t 2
1 t/10
10
=
e
+
.
2
10 + t
Therefore for t > 0,
dP (T > t)
dt
1
10
1
= et/10
2
10
MATH 230/STAT 230: Homework 8
Due on 11/10, in class
Read Section 4.2, and Section 4.4
Problem 1. Assume that calls arrive at a call centre according to a Poisson arrival process
with a rate of 15 calls per hour. For 0 s < t, let N (s, t] denote the numbe
MATH 230/STAT 230: Homework 7
Due on 10/29, in class
Read Section 3.4, Section 3.5, and Section 4.1
Problem 1. Suppose X takes values in (0, 1) and has a density
f (x) = cx2 (1 x)2
x (0, 1)
for some c > 0.
(a) Find c.
(b) Find E(X).
(c) Find Var(X).
Probl
MATH 230/STAT 230: Homework 6
Due on 10/22, in class
Read Section 3.3-Section 3.4
1
Problem 1. Let A1 , A2 , and A3 be events with probabilities 1 , 4 , and 1 respectively. Let
5
3
N be the number of these events that occur. For example, N = 2 if two of A
MATH 230/STAT 230: Homework 5
Due on 10/15, in class
Read Section 3.1-Section 3.2
Problem 1. Let us assume that we have n + m independent Bernoulli (p) trials. For any
r > 0, let Sr be the number of successes in the rst r trials, Tr the number of successe
MATH 230/STAT 230: Homework 4
Due on 09/29, in class
Read Chapter 2 (excluding Section 2.3), Appendix 1, and Section 3.1 (up to page 148)
1
Problem 1. Let S be the number of successes in 25 independent trials with probability 10
of success on each trial.
MATH 230/STAT 230: Homework 3
Due on 09/17, in class
Read Section 1.5-Section 1.6, and Section 2.1
Problem 1. A biased coin lands head with probability 2/3. It is tossed three times.
(a) Given that there is at least one head in three tosses, what is the c
MATH 230/STAT 230: Homework 2
Due on 09/10, in class
Read Section 1.1-Section 1.5
Problem 1. Show that for any collection of events cfw_Ai n+1 ,
i=1
n+1
n
i=1
n
Ai + P(An+1 ) P
Ai = P
P
i=1
(Ai An+1 ) .
(1)
i=1
Problem 2. (i) For two events A and B show t
MATH 230/STA 230: Homework 1
Problem 1. Show that (i)
n
1
j = n(n + 1),
2
j=1
and
(ii)
n
j2 =
j=1
n(n + 1)(2n + 1)
.
6
Hint: (i) Notice that 1 + n = n + 1, 2 + (n 1) = n + 1, 3 + (n 2) = n + 1 and so on.
(ii) Note (j + 1)3 j 3 = 1 + 3j + 3j 2 .
Problem 2.
Name
Math 135 Midterm 1 Solutions
Math 135.01 Probability
Feb. 13, 2007
There are ve questions on this test. Partial credit will be given on questions 2 through 5. DO use
calculators if you need them. You are allowed one sheet both sides with whatever you
Homework 6 Solution
3.2 14. Let Ii be the indicator that the elevator will stop on the ith oor, which is
equivalent to say that there are at least one person who will get out on the ith oor.
Therefore, the expectation of Ii is
9
10
E(Ii ) = P (Ii = 1) = 1
Homework 8 Solution
1
4.4 4. The density of X is fX (x) = 2 I(x (1, 1). The function Y = g(X) = X 2 is
not monotone on [1, 1], but is monotone on [1, 0] and on (0, 1]. On [1, 0], g 1 (y) =
y; on [0, 1], g 1 (y) = y. The range of Y is [0, 1). Using the ch
Chapter 3.1,3.3,3.4
CLT
Markovs and Chebyshevs Inequalities
For any random variable X 0 and constant a > 0 then
Lecture 8: Using the LLN and CLT, Moments of
Distributions
Markovs Inequality:
P(X a)
Sta230/Mth230
E (X )
a
Chebyshevs Inequality:
Colin Rund
Chapter 3.1,3.3,3.4
A little more E (X )
Practice Problem - Skewness of Bernoulli Random Variable
Let X Bern(p) We have shown that
Lecture 7: Joint Distributions and the Law of Large
Numbers
E (X ) = p
Var (X ) = p(1 p)
Sta230/Mth230
Find the Skewness of
Chapter 3.1-3.3
Random Variables
Random Variables
We have been using them for a while now in a variety of forms but it is
good to explicitly dene what we mean
Lecture 6: E (X ), Var (X ), & Cov (X , Y )
Random Variable
A real-valued function on the sample
Solutions to HWK 1
7. The probability is 3/4, since they win unless the other team wins the last two games.
10. We can get 1 + 2 + 6, 1 + 3 + 5, 2 + 3 + 4 in 3! = 6 ways, 1 + 4 + 4, 2 + 2 + 5 in 3
ways, and 3 + 3 + 3 in 1 way so the number of possibilitie
Answers to Homework 4 in Math 230, Spring 2016
3.1. A friend flips two coins and tells you that at least one is Heads. Given this information,
what is the probability that the first coin is Heads?
A =at least one is Heads = cfw_(H, T ), (T, H), (H, H) and
Answers to Homework 3 in Math 230, Fall 2013
2.39. The probability of a three of a kind in poker is approximately 1/50. Use the Poisson
approximation to compute the probability you will get at least one three of a kind if you
play 20 hands of poker.
Answe