DEPARTMENT OF MATHEMATICS
SYLLABUS
Course # & Name:
MAT 135A: Probability Theory
Recommended Text(s) & Price:
Prepared by:
C. Tracy and A.
Soshnikov
Lecture(s)
Probability and Stochastic Processes,
Frederick Solomon, 1st Edition, Prentice
Hall, ISBN-10: 0
When was alphabetic writing adopted by the Greeks?
A.ClassicalPeriod
B.BronzeAge
C.ArchaicPeriod
D.DarkAge
E.HellenisticPeriod
Reset Selection
Answer
Key: C
Feedback: Incorrect. The alphabet was introduced in the Archaic
Period.
Question 2 of 13
Which god
Q1
Plotting alice and bob's scores on a grid, the problem can be visualized as counting the number of paths from (3, 2) to the winning state of Alice or Bob, as depicted
in the following diagram.
A
A
(3, 5) (4, 5)
(3, 4) (4, 4) (5, 4) B
(3, 3) (4, 3) (5,
MAT 135A 2014-2015 PRACTICE MIDTERM 1
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assessed a 20 points deduction on their final scores.
Please, write your answers using a pen, not a pencil.
All your answers must be written
MAT 135A 2014-2015 PRACTICE FINAL
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assessed a 20 points deduction on their final scores.
Please, write your answers using a pen, not a pencil.
All your answers must be written ins
Real Analysis
Math 125A, Fall 2012
Sample Final Questions
1. Define f : R R by
x3
.
1 + x2
Show that f is continuous on R. Is f uniformly continuous on R?
f (x) =
Solution.
To simplify the inequalities a bit, we write
x
x3
=x
.
2
1+x
1 + x2
For x, y R, w
Problem 1
Suppose that f is a function mapping R into R and there exists > 0 such that
|f (x) f (y)| < |x y|1+ for all x and y in R. Show first that f is differentiable
at every point x in R. Then show f is a constant function.
Proof. Fix x R. Let > 0. De
Erwin Macalalad
Math 150A
Homework 4
August 28 2016
Homework 4
5.1.3 Is On isomorphic to the product group SOn cfw_ I ?
Let On be an orthogonal group. SOn is a subgroup of On with determinant 1.I = At A
is an orthogonal group as well. Since SOn is a subgr
1. Find a E R and b 75 0 such that
lim N 112012 = I).
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7).:1
Fully explain your answer.
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MATH 125B, HW2, Solution
5.3.2(a) Change the variable t = x, and dx = 2tdt. then
Z 4
Z 2
Z
f ( x)dx =
f (t)2tdt = 2
1
1
2
f (t)tdt = 12
1
3
5.3.2(b) Change the variable t = x12 , and dx = 21 t 2 dt. then
Z 1
Z
Z 1
3
1 3
1 2
1
2
f (t) t
f (t)t 2 dt.
dt =
MATH 125B, HW1, Solution
5.1.1(b) Since f (x) = 3 x2 is decreasing on the interval [0, 2], then
1 1
1
U (f, P ) = f ( )( 0) + f (1)(1 ) + f (2)(2 1);
2 2
2
1
1
1
L(f, P ) = f (0)( 0) + f ( )(1 ) + f (1)(2 1).
2
2
2
Graph and use geometric interpretation o
Chapter 3 Summary
Expectation of Absolutely Continuous Random Variables
Expectation of Discrete Random Variables
If X is an absolutely continuous random variable with
density fX , then
Z
E(X) =
xfX (x) dx .
If X is a discrete random variable, then
X
E(