This document includes three practice questions/examples
for Quiz 3/ midterm (to be held Nov. 8,
Oct
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3
2016, in-class).
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list
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of questions for midterm 2 (from Montgomery 8th ed.): 6.1, 6.3, 7.1, 7.21, 8.11, 8.14 (ANOVA
table
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CHEM 2252
CHEM 2252: Treatment of
analytical data, Chromatography
Questions/Main Ideas:
What is Mean?
What is Mode, Median, and
Standard deviation?
What is range?
What is the formula to calculate
standard deviation?
Formula to calculate Variance
Formula t
CHEM 2252
CHEM 2252: Analytical
Chemistry, Types of Analysis.
Questions/Main Ideas:
What are the branches of
chemistry?
What is analytical chemistry?
What are the types of analysis?
Qualitative Analysis
Quantitative Analysis
Characterization Analysis
Fund
1
Math 302 Final Exam
Section: 101
Instructor: Ed Perkins
Duration: 2.5 hours .
Instructions:
Write your name and student ID on every page.
This examination contains eight questions worth a total of 102 points.
Write each answer very clearly below the
April 2005
MATH 302 Section 201
Name:
1
1) A poker hand consist of 5 cards out of a deck of 52. a) What is the probability of a Full House, that is, three cards of one denomination and two cards of a second denomination. [5] b) What is the probability of
Math 302 Solutions to assignment 6
1. We have that
a
cx4 =
1=
0
and
ca5
,
5
a
cx5 =
1 = E[X] =
0
ca6
,
6
so we deduce that a = 6/5 and c = 5/a5 . Hence
a
ca7
5a2
36
2
E[X ] =
cx6 =
=
=
,
7
7
35
0
so Var(X) = 1/35.
2. (a) This is the probability that a un
Math 302 Solutions to Assignment 5
1. (a) Write p for the number p = P (X = 1). Then EX = p and EX 2 = p so V ar(X) = pp2 .
The equality EX = 3V ar(X) then turns to be p = 3p 3p2 and so p = 2/3, hence
P (X = 0) = 1/3.
(b) Write n, p for the parameters of
Math 302 Assignment 8
1. For any a 0 we have
ay
FX/Y (a) = P (X/Y a) =
1 2 e
0
0
= 1
2 e(2 +1 a)y dy = 1
1 x 2 x
e
2
2 +1
=
1
2 +1
dxdy =
(1 e1 ay )2 e2 y
0
2
.
2 + 1 a
0
So P (X < Y ) = FX/Y (1) = 1
.
2.
P (|X Y | L/4) =
L/4 L
2
(L/2)
0
L/2
= 1/2 +
Math 302 Solutions to assignment 4
1. (a) We have that
P (X > m + n, X > m)
(1 p)m+n
=
= (1 p)n = P (X > n) .
P (X > m)
(1 p)m
(b) We have that
E[rX ] =
(1 p)k1 prk = rp
[r(1 p)]j =
j=0
k=1
rp
,
1 r(1 p)
where in the last equality we used the geometric su
Math 302 solutions to assignment 3
1. Let A be the event that the two dice land on dierent numbers and B the event that one
of the dice is 6. Then P (A) = 30/36 = 5/6 because there are 30 pairs of dierent numbers
in cfw_1, . . . , 6. Also, P (A B) = 10/36
Math 302 solutions to Assignment 10
1. (25 pts) Let X1 , X2 , . . . , X100 be i.i.d. random variables each equal 2 with probability .8 and
1 with probability .2. And write R = X1 X2 X100 .
(a) Use Chebychevs inequality to bound the probability P (275 R 28
1
Math 302 Solutions to practice nal 1
1. (a) A, B, C are independent events if all of the following occur:
P (A B) = P (A)P (B),
P (B C) = P (B)P (C),
P (A C) = P (A)P (C),
P (A B C) = P (A)P (B)P (C),
i. We have that P (X Y = 0) = 1/2 (because this only
1
Math 302 practice midterm
Instructor: Asaf Nachmias
Duration: 50 minutes.
Instructions:
Write your name and student ID on every page.
This examination contains four questions with weight 25 points each.
Write each answer very clearly below the corres
1
Math 302 midterm solutions
1. (a) (9 pts) Carefully dene (with a formula) what it means for two events A and B to
be independent.
P (A B) = P (A) P (B) .
(b) (9 pts) How many ways are there to arrange 3 novels, 2 math books and 1 chemistry
book in a she
1
Math 302 Practice Final 1
Instructor: Asaf Nachmias
Duration: 2.5 hours .
Instructions:
Write your name and student ID on every page.
This examination contains six questions with weight 17 points each (102 points total).
Write each answer very clearl
1
Math 302 Practice Final 2
Instructor: Asaf Nachmias, section 102
Duration: 2.5 hours .
Instructions:
Write your name and student ID on every page.
This examination contains six questions with weight 17 points each (102 points total).
Write each answe
1
Math 302 solutions for practice midterm
1. (a) Carefully dene (with a formula) what it means for two discrete r.v.s X and Y to
be independent.
For any two numbers a1 , a2 we have P (X = a1 , Y = a2 ) = P (X = a1 )P (Y = a2 ).
(b) How many quadruples (w,
1
Math 302 solutions to practice nal 2
1. (a) Let X be a binomial r.v. with parameters n and p. Write the probability mass
function of X.
For any k = 0, 1, . . . , n we have P (X = k) =
(n )
k
pk (1 p)nk .
(b) Suppose that X is a r.v. receiving the values
Math 302 solutions to assignment 9
1. (20 pts) In order to calculate E[X], E[Y ] we calculate the marginal densities rst.
1
fX (x) =
f (x, y)dy = ln(x) = ln(x1 )
0 x 1.
x
fY (y) =
y
0 y 1.
f (x, y)dx = 1
0
[
]1
1
So E[Y ] = 1/2 and E[X] = 0 x ln(x) = x2
1
Math 302 midterm
Instructor: Asaf Nachmias
Duration: 50 minutes.
Instructions:
Write your name and student ID on every page.
This examination contains four questions. The total number of points is 101.
Write each answer very clearly below the corresp
Application of Determinant to Systems:
Cramer's Rule
We have seen that determinant may be useful in finding the inverse of a nonsingular
matrix. We can use these findings in solving linear systems for which the matrix
coefficient is nonsingular (or invert
Computation of Eigenvectors
Let A be a square matrix of order n and
eigenvector of A associated to
one of its eigenvalues. Let X be an
. We must have
This is a linear system for which the matrix coefficient is
. Since the zerovector is a solution, the sys
Determinants of Matrices of Higher Order
As we said before, the idea is to assume that previous properties satisfied by the
determinant of matrices of order 2, are still valid in general. In other words, we
assume:
1.
Any matrix A and its transpose have t
Determinant and Inverse of Matrices
Finding the inverse of a matrix is very important in many areas of science. For
example, decrypting a coded message uses the inverse of a matrix. Determinant may
be used to answer this problem. Indeed, letA be a square
Computation of Eigenvalues
For a square matrix A of order n, the number
exists a non-zero vector C such that
is an eigenvalue if and only if there
Using the matrix multiplication properties, we obtain
This is a linear system for which the matrix coefficie
Eigenvalues and Eigenvectors: An
Introduction
The eigenvalue problem is a problem of considerable theoretical interest and wideranging application. For example, this problem is crucial in solving systems of
differential equations, analyzing population gro
IntroductiontoDeterminants
For any square matrix of order 2, we have found a necessary and sufficient condition
for invertibility. Indeed, consider the matrix
The matrix A is invertible if and only if
. We called this number
the determinant of A. It is cl
Systems of Linear Equations: Gaussian
Elimination
It is quite hard to solve non-linear systems of equations, while linear systems are quite
easy to study. There are numerical techniques which help to approximate nonlinear
systems with linear ones in the h
The Case of Complex Eigenvalues
First let us convince ourselves that there exist matrices with complex eigenvalues.
Example. Consider the matrix
The characteristic equation is given by
This quadratic equation has complex roots given by
Therefore the matri
SYSTEMS OF EQUATIONS in TWO VARIABLES
A system of equations is a collection of two or more equations with the same set of
unknowns. In solving a system of equations, we try to find values for each of the
unknowns that will satisfy every equation in the sy