Statistics 2060 Midterm Exam
Date: Thursday, March 4th: 16:30 18:30
With the solutions
Instructor: Dr. Ammar Sarhan
Name: _
Solution
_
Student ID # : _
This midterm exam has 8 pages with total of 66 marks. The number of points allocated to
each portion of
Stat 2060 - Joint Probability Distribution
Lecture Notes Chapter 5
Dr. Ammar Sarhan
Department of Mathematics & Statistics,
Dalhousie University
October 23, 2016
Dr. Ammar Sarhan
Joint Probability Distribution
October 23, 2016
1 / 21
5.1 Joint distributio
Introduction to Probability and
Statistics
Chapter 4
Ammar M. Sarhan,
asarhan@mathstat.dal.ca
Department of Mathematics and Statistics,
Dalhousie University
Chapter 4
Continuous Random
Variables and
Probability Distributions
Dr. Ammar Sarhan
2
4.1 Continu
Math 2120
Test 1
Monday, May 16, 2016
You only need something to write with. No formula sheets of any kind.
If I suspect you of cheating you will have your test taken from you.
Make sure there is a seat between you and the people on either side of
you.
CHAPTER 6. MATRIX ALGEBRA
6.4
44
Solving Systems by Reducing Matrices
Example 6.4.1. The admission fee at a small fair is $1.50 for children and
$4.00 for adults. On a certain day, 2200 people enter the fair and $5,050 is
collected. How many children and
CHAPTER 11. DIFFERENTIATION
11.5
97
The Chain Rule
Given functions f and g, the composition of f and g, denoted f
(f
g, is given by
g)(x) = f (g(x).
Computing the value of the composite function f g at x is a two-step process:
we first substitute x into g
Chapter 8
Introduction to Probability
and Statistics
8.1
Basic Counting Principle and Permuations
Example 8.1.1. Jessie has 5 tops, 3 pairs of pants, and 2 pairs of shoes. In
how many ways can she get dressed?
Solution: We can systematically list her opti
CHAPTER 5. MATHEMATICS OF FINANCE
5.4
20
Annuities
An annuity is a finite sequence of payments made at fixed periods of time over a
given interval. We consider periods of time that are always of equal length, and
this length is called the payment period.
Chapter 5
Mathematics of Finance
5.1
Compound Interest
Under compound interest, interest earned at the end of each period is added to
the existing principal, so that it too earns interest in the next period.
Example 5.1.1. Suppose I invest $100 in a savin
CHAPTER 6. MATRIX ALGEBRA
6.3
40
Matrix Multiplication
The definition of matrix multiplication is more involved than the definition of
matrix addition. Matrix addition is simply addition of the corresponding entries,
but it turns out that multiplication o
CHAPTER 5. MATHEMATICS OF FINANCE
5.2
15
Present Value
Example 5.2.1. Suppose a first year student at Dalhousie wants to purchase
a car during her fourth year, and has decided that she will need $3000 saved in
order to make a down payment. She has found a
CHAPTER 5. MATHEMATICS OF FINANCE
5.5
28
Amortization of Loans
Suppose that a bank lends a borrower $1,000, and charges interest at a rate
of 12% compounded monthly. This loan is to be repaid in five monthly installments, paid at the end of each month. Wh
CHAPTER 11. DIFFERENTIATION
11.4
96
The Product Rule and the Quotient Rule
The constant factor rule and the sum and dierence rule worked pretty much the
way we would expect them to naively. For products and quotients of functions,
things are not so straig
CHAPTER 10. LIMITS AND CONTINUITY
10.3
84
Continuity
Intuitively, a function is continuous if its graph can be drawn without lifting
the pencil o of the page. How can we state this mathematically?
Definition 10.3.1. A function f is continuous at a if and
CHAPTER 10. LIMITS AND CONTINUITY
10.4
88
Continuity Applied to Inequalities
We are often interested in where a function is positive or negative. For example,
we may wish to know where the function f (x) = x2 + 3x
4 is positive and
negative. Since factori
CHAPTER 6. MATRIX ALGEBRA
6.2
35
Matrix Addition and Scalar Multiplication
Example 6.2.1. Consider a guitar dealer who sells two models of Gibson guitars, the Les Paul and the SG. Each model is available in three tiers; standard,
classic, and custom. The
Chapter 10
Limits and Continuity
10.1
Limits
Definition 10.1.1. For a function f and a real number a, we say that the limit
of f (x) as x approaches a is L, and write
lim f (x) = L,
x!a
if for all x sufficiently close to a (but not equal to a), the value
Chapter 4
Exponential and
Logarithmic Functions
4.1
Exponential Functions
Definition 4.1.1. The function f defined by
f (x) := bx ,
where b > 0, b 6= 1, and the exponent x is any real number, is called an
exponential function with base b.
Example 4.1.1. C
Chapter 6
Matrix Algebra
6.1
Matrices
Matrices are rectangular arrays of numbers.
Example 6.1.1. The matrix
2
6 1
A=4
3
3
1 27
5
4 1
has two rows and three columns, so we say that it has size 2 3 (the number
of rows always comes first when we state the si
CHAPTER 10. LIMITS AND CONTINUITY
10.2
80
Limits (Continued)
Consider the function f (x) = x1 , pictured below.
y
4
y = f (x)
3
2
1
4
3
2
1
1
2
3
4
x
1
2
3
4
The limit
lim
x!0
1
x
has the form 10 . Looking at the graph, the behaviour appears very dierent
CHAPTER 13. CURVE SKETCHING
13.2
111
Absolute Extrema on a Closed Interval
Definition 13.2.1. A function f has an absolute maximum on the interval I if
there is a point a in I with f (a)
f (x) for all x in I. The absolute maximum
is f (a). In other words,
CHAPTER 5. MATHEMATICS OF FINANCE
5.6
30
Perpetuities
An infinite sequence of payments is called a perpetuity.
Example 5.6.1. A school wants to set up a scholarship that will pay out
$5,000 per year. What lump sum will guarantee that this payment can be m
CHAPTER 5. MATHEMATICS OF FINANCE
5.3
18
Interest Compounded Continuously
Example 5.3.1. What is the value of a $1000 investment after 1 year if it is
invested at a nomial rate of 5% compounded
(a) annually?
S = 1000(1 + 0.05)1 = 1050
(b) quarterly?
S = 1
CHAPTER 11. DIFFERENTIATION
11.2
94
Rules for Dierentiation
Having computed the derivatives of several functions using the limit definition,
we realize that we may want a faster way to compute derivatives. In this section
we start to build up some rules f
CHAPTER 6. MATRIX ALGEBRA
6.6
55
Inverses
Here we use another method of solving linear equations which works for systems
of n linear equations in n variables. First note that a system of linear equations
can be written in matrix form. For example, the sys
Chapter 13
Curve Sketching
13.1
Relative Extrema
Definition 13.1.1. A function f is said to be increasing on an interval I when
for any two numbers a and b in I, if a < b then f (a) < f (b). This means that
the function is rising from left to right.
A fun
Chapter 11
Dierentiation
11.1
The Derivative
y
4
Q = (xQ , yQ )
2
P = (xP , yP )
2
1
1
2
3
4
x
5
2
Definition 11.1.1. For points P and Q on a curve, the secant line is the line
passing through P and Q
The slope of the secant line is the average rate of ch
Chapter 7
Linear Programming
7.2
Linear Programming
This section of the notes uses large parts from Section 7.1 of the textbook
as well. Make sure to read both Sections 7.1 and 7.2 of the textbook. We
start by working through an example completely, and th