CHEMISTRY 102B
Practice Hour Exam I
Spring 2015
Dr. D. DeCoste
Name _
Signature _
T.A. _
This exam contains 17 questions on 5 numbered pages. Check now to make sure you
have a complete exam. You have one hour and thirty minutes to complete the exam.
Deter
1. I choose a real number uniformly at random in the interval [2, 6], and call it X .
(a) Find the CDF of X, FX (x).
(b) Find EX .
2. Let X be a continuous random variable with the following PDF
f X ( x) =
ce4x
0
x0
otherwise
where c is a positive constan
CHEMISTRY 102B
Practice Hour Exam II
Spring 2015
Dr. D. DeCoste
Name _
Signature _
T.A. _
This exam contains 22 questions on 8 numbered pages. Check now to make sure you have a
complete exam. You have one hour and thirty minutes to complete the exam. Dete
CHAPTER 1. BASIC CONCEPTS
68
1.5
End of Chapter Problems
1. Suppose that the universal set S is dened as S = cfw_1, 2, , 10 and A = cfw_1, 2, 3,
B = cfw_X S : 2 X 7, and C = cfw_7, 8, 9, 10.
(a) Find A B
(b) Find (A C ) B
(c) Find A (B C )
(d) Do A, B, an
Chapter 3
Discrete Random Variables
3.1
3.1.1
Basic Concepts
Random Variables
In general, to analyze random experiments, we usually focus on some numerical aspects of the
experiment. For example, in a soccer game we may be interested in the number of goal
1
Problems
1. Suppose that the universal set S is dened as S = cfw_1, 2, , 10 and A = cfw_1, 2, 3,
B = cfw_X S : 2 X 7, and C = cfw_7, 8, 9, 10.
(a) Find A B
(b) Find (A C ) B
(c) Find A (B C )
(d) Do A, B, and C form a partition of S ?
Solution:
(a)
A B
1
Problems
1. Let X be a discrete random variable with the following PMF
1
for x = 0
2
1
for x = 1
3
1
PX (x) =
for x = 2
6
0
otherwise
(a) Find RX , the range of the random variable X .
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|X < 2)
1. I choose a real number uniformly at random in the interval [2, 6], and call it X .
(a) Find the CDF of X, FX (x).
(b) Find EX .
Solution:
(a)
We saw that all individual points have probability 0; i.e.,P (X = x) = 0 for all
x in uniform distribution. Al
1. Let X be a random variable with the following CDF:
0
x
FX (x) =
for x < 0
for 0 x <
x+
1
1
2
for
1
4
x<
for x
1
4
1
2
1
2
(a) Plot FX (x) and explain why X is a mixed random variable.
1
(b) Find P (X 3 ).
(c) Find P (X 1 ).
4
(d) Write CDF of X in th
Solutions Manual for
Probability and Random Processes for
Electrical and Computer Engineers
John A. Gubner
University of WisconsinMadison
File Generated July 13, 2007
CHAPTER 1
Problem Solutions
1. = cfw_1, 2, 3, 4, 5, 6.
2. = cfw_0, 1, 2, . . . , 24, 25.
1
Problems
1. Let X be a discrete random variable with the following PMF
1
for x = 0
2
1
for x = 1
3
1
PX (x) =
for x = 2
6
0
otherwise
(a) Find RX , the range of the random variable X .
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|X < 2)
1. Let X be a random variable with the following CDF:
0
x
FX (x) =
for x < 0
for 0 x <
x+
1
1
2
for
1
4
x<
for x
1
4
1
2
1
2
(a) Plot FX (x) and explain why X is a mixed random variable.
1
(b) Find P (X 3 ).
(c) Find P (X 1 ).
4
(d) Write CDF of X in th
1
Problems
1. Let X be a discrete random variable with the following PMF
1
for x = 0
2
1
for x = 1
3
1
PX (x) =
for x = 2
6
0
otherwise
(a) Find RX , the range of the random variable X .
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|X < 2)
1. Consider two random variables X and Y with joint PMF given in Table 1.
Table 1: Joint PMF of X and Y in Problem 1
Y =1 Y =2
X=1
1
3
1
12
X=2
1
6
0
X=4
1
12
1
3
(a) Find P (X 2, Y > 1).
(b) Find the marginal PMFs of X and Y .
(c) Find P (Y = 2|X = 1).
(
1
Problems
1. A coee shop has 4 dierent types of coee. You can order your coee in a small,
medium, or large cup. You can also choose whether you want to add cream, sugar,
or milk. In how many ways can you order your coee?
Solution:
We can use the multipli
1. Let X be a discrete random variable with the following PMF:
1
for x = 0
2
1
for x = 1
3
1
PX (x) =
for x = 2
6
0
otherwise
(a) Find RX , the range of the random variable X .
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|X < 2)
2. Let X
1. I choose a real number uniformly at random in the interval [2, 6], and call it X .
(a) Find the CDF of X, FX (x).
(b) Find EX .
Solution:
(a)
We saw that all individual points have probability 0; i.e.,P (X = x) = 0 for all
x in uniform distribution. Al
1
Problems
1. I choose a real number uniformly at random in the interval [2, 6], and call it X .
(a) Find the CDF of X, FX (x).
(b) Find EX .
2. Let X be a continuous random variable with the following PDF
f X ( x) =
ce4x
0
x0
otherwise
where c is a posit
Lecture 19:
Building Atoms and Molecules
+e
r
n=3
n=2
n=1
+e
+e
r
yeven
Lecture 19, p 1
Today
Atomic Configurations
States in atoms with many electrons
filled according to the Pauli exclusion principle
Molecular Wave Functions: origins of covalent bonds
Lecture 11
Applying Boltzmann Statistics
Elasticity of a Polymer
Heat capacities
CV of molecules for real !
When equipartition fails
Not on
midterm.
Planck Distribution of Electromagnetic Radiation
Reading for this Lecture:
Elements Ch 9
Reading for Le
213 Midterm coming up
Monday April 9 @ 7 pm (conflict exam @ 5:15pm)
Covers:
Lectures 1-12 (not including thermal radiation)
HW 1-4
Discussion 1-4
Labs 1-2
Review Session
Sunday April 8, 3-5 PM, 141 Loomis
HW 4 is not due until Thursday, April 12 at 8 am,
Lecture 12
Examples and Problems
p(h)/p(0)
1.0
0.8
<h>=kT/mg
0.6
0.4
0.2
0.0
0
1
2
3
4
5
mgh/kT
Lecture 12, p1
Boltzmann Distribution
If we have a system that is coupled to a heat reservoir at temperature T:
The entropy of the reservoir decreases when th
Lecture 13
Heat Engines
Thermodynamic processes and entropy
Thermodynamic cycles
Extracting work from heat
- How do we define engine efficiency?
- Carnot cycle: the best possible efficiency
Reading for this Lecture:
Elements Ch 4D-F
Reading for Lecture
Lecture 14
Heat Pumps, Refrigerators, and
Bricks!
Pumping Heat: Heat pumps and Refrigerators
Available Work and Free Energy
Work from Hot and Cold Bricks
Reading for this Lecture:
Elements Ch 10
Reading for Lecture 16:
Elements Ch 11
Lecture 14, p 1
Ru
Miscellaneous Notes
The end is near dont get behind.
All Excuses must be taken to 233 Loomis
before 4:15, Monday, May 1.
The PHYS 213 final exam times are
* 8-10 AM, Monday, May 7
* 8-10 AM, Tuesday, May 8
and
* 1:30-3:30 PM, Friday, May 11.
The deadli
Lecture 15
Heat Engines
Review & Examples
p
pb
b
Hot reservoir at Th
Heat
leak
c
pa
a
adiabats
V2
Heat
pump
W
Qc
Cold reservoir at Tc
d
V1
Qh
V
Lecture 15, p 1
Review
Entropy in Macroscopic Systems
Traditional thermodynamic entropy: S = k lnW = ks
We want
Lecture 8
The Second Law of Thermodynamics;
Energy Exchange
The second law of thermodynamics
Statistics of energy exchange
General definition of temperature
Why heat flows from hot to cold
Reading for this Lecture:
Elements Ch 7
Reading for Lecture 10