1
Stochastic Processes,
Theory for Applications
Solutions to Selected Exercises
R.G.Gallager
October 5, 2014
The complete set of solutions is available to instructors teaching this course. Contact
Cambridge Press at www.Cambridge.org.
The solutions here o
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Figure 1.2 Correlations between lattice observations Z(si) and the average of the nearest neighbors for lattice
arrangements shown in Figure 1.1.
Here, Z denotes the attribute we observe, for example, yield, conce
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Example 1.1 A drug company is interested in testing the uniformity of its buffered aspirin tablets for quality control
purposes. A random sample of three batches of aspirin tablets is collected from each of the co
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Similarly, in the longitudinal case, the observations from different subjects can be collected into a single vector,
and model (1.1) can be written as
Y=Xl+e,
(1.3)
where Xl is a stacking of X1, X2, , Xn, and e is
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CHAPTER 1
Introduction
1.1 The Need for Spatial Analysis
Statistical methods for spatial data analysis play an ever increasing role in the toolbox of the statistician, scientist,
and practitioner. Over the years,
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CHAPTER 1
Introduction
1.1 The Need for Spatial Analysis
Statistical methods for spatial data analysis play an ever increasing role in the toolbox of the statistician, scientist,
and practitioner. Over the years,
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cation mechanism. The computational ramifications of a dense matrix are formidable in their own right. For
example, computing in the subject-by-subject form of a longitudinal model requires inversion of Vi, a matr
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continuous or discrete, has no bearing on whether the data are geostatistical, or not.
Example 1.3 Consider measuring air temperature. Air temperature could, at least theoretically, be recorded at any
location in
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Figure 1.1 Simulated spatial arrangements on a 1010 lattice. Independent draws from a G(5, 1) distribution
assigned completely at random are shown in panel a. The same data were then rearranged to create arrangeme
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correlated, these data fall within the realm of traditional random sampling. We consider the vectors Yi=[Yi(t1), ,
Yi(tni)] as independent random vectors because patients were selected at random. A statistical mod
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models with nugget effect from consideration and reflects the sentiment that only the study of mean square
continuous processes is worthwhile.
Mean square continuity by itself does not convey much about the smoo
Interests on Transactions
Chapter 10 13
PV & FV of Annuities
An engineer deposits P1,000 in a savings account at the
end of each year for 5 years. How much money can he
withdraw at the end of 5 years if the bank pays interest
at the rate of 6% p.a. compou
Doing Business
Chapter 7 & 8
1
Get Discounts
on purchases
Buy on credit
or pay in cash
for more
discounts
Put Markup
on sales
Gain Profit
2
Payment = List Price Trade Discounts Cash Discounts + Freight
(1) Trade Discounts: Single
Chain
(2) Cash Discounts:
Practice Problems Set 1
Time Value of Money
Problem 1
1. A loan of Php 500,000 is due 5 years from
today. (a) The borrower wants to make annual
deposits at the end of each year into a sinking
fund that will earn interest at an annual rate of
8%, compounde
Practice Problems Set 2
Time Value of Money
Problem 1
Two years ago, you bought a new sports car
with a P1.2 million bank loan at an interest
rate of 9 % per year compounded monthly.
The loan was to be paid in 48 months. After
24 months of payment, you d
Practice Problems
Discounts Pricing 1
Practice Problem # 1
Kikos Bake Shop bakes 800 loaves of
bread per week at a cost of P70 each.
A mark up of 70% on cost is added in
setting up the selling price. Based on
experience, Kiko knows that 20 % of
the bread
Single Transactions
Time Value of Money
Chapter 7 8 , 10 13
Interest on transactions is an amount that is charged for the use of somebody elses money
You borrow P30,000, to be paid one year after , at an interest rate of 8% per
year.
P
MV
One year
P = Pr
SBEP
Applied Math
Contact Information
Manuel A. Tenmatay
[email protected]
0917- 837-4570
3
Course Title:
Applied
Mathematics
Textbook:
Practical Business
Math Procedures
(11th edition)
by Jeffrey Slater
(McGraw Hill)
Course Contents:
Application of mat
Practice Problems
Breakeven Point
1
1. Master Yu is planning to get a franchise to sell a unique recipe of turon. The
franchise fee is P 120,000 to be paid in 24 equal monthly installments at zero
interest. To operate, he needs two crew members with a com
Applied Math
Multi-Product Breakeven Point
1
Contribution Margin = Amount remaining after variable costs
are deducted from sales amount
Contribution Margin Ratio = Contribution Margin expressed
as a percentage of sales
10%
Overall CM Ratio = Total CM / To
Stock Name
Price
Increment
Stocks
from
DAY 1
Price
Buy
Sell
%inc/ dec
DAY2
Price
Buy
Sell
%inc/ dec
DAY3
Price
Buy
Sell
%inc/ dec
DAY4
Price
Buy
Sell
%inc/ dec
DAY5
Price
Buy
Sell
%inc/ dec
DAY6
Price
Buy
Sell
%inc/ dec
DAY7
Price
Buy
Sell
%inc/ dec
DAY8
First Question on slide 17:
Try compounding P10,000 annually, semianually, quarterly, monthly, daily at a
rate of 5% per annum. What are the effective rates here?
Period
No of
periods (N)
Interest
rate per
period (R)
Principal =
10000
Interest
(FV-PV)
APY
Applied Math
Chapter 8: Markups and Markdowns Perishables and Breakeven Analysis
1
Formula
Markups Based on Cost
Notes
Selling Price (P) = Cost (C) + Markup (M)
2
a.
P = C * (1 + Percent Markup on Cost (RC)
a. Calculating Selling Price when you know
Cost
CHAPTER 6 Problem 7
Assume that Mei has $100 per month to divide between dinners at a Chinese restaurant and evenings at
Zanzibar, a local pub. Assume that going to Zanzibar costs $20 and eating at the Chinese restaurant costs
$10. Suppose Mei spends two
Chapter 3: Demand, Supply, and Equilibrium Prices
Chapter 4: Demand and Supply Applications
The following graph represents the market for wheat. The equilibrium price is $20 per bushel
and the equilibrium quantity is 14 million bushels.
o
Explain what wil