Notesale: Turn your study into money

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CHAPTER ONE

1
...
All linear
functions have a constant derivative because the tangent at every point is the line itself
...
But non-linear functions
have derivatives whose value depends on the point x at which it is measured
...
It has value
0 at the point x=−1, a positive value when x>−1, and a negative value when x<−1
...

DEFINITIONS
We can define the derivative of a function f as:
𝑓´(𝑥) = lim (𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥) )
Δx↓0
∆x
This is if we take the slope of the chord between two points, a distance ∆𝑥 apart and see what
happens as the two points get closer and closer
...
The slope of the chord
is:
𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥
∆𝑥
If a tangent line is drawn and its slope is the derivative 𝑓 ′ (𝑥)
...

The Second derivative is the derivative of the derivative and is given by:𝑓 ′′ 𝑥 = 𝑙𝑖𝑚
∆𝑥 ↓ 0


𝑓 ′ 𝑥 + ∆𝑥) − 𝑓 ′ (𝑥)
}
∆𝑥

If we differentiate a function m times, we denote the m th derivative by f (m) (x)

We can use the alternative notation 𝑑𝑓 𝑑𝑥 for the first derivative 𝑓’ (𝑥) and dm f also stands
for
d xm
the mth derivative


The differential operator d is associated with the definition of the derivative
...
The differential operator may be used for example to describe
the dynamics of financial asset returns in continuous time
...
Obtain the derivative of the following:
(i) X3
(ii) 2X5
2
...
1 RULES FOR DIFFERENTIATION
A number of simple rules for differentiation may be used to calculate the derivatives of
certain functions from first principles
...
Power
...
Exponential
...
Logarithm
The derivative of ln x is x-1
𝑑

𝑑𝑥(ln x) =

1

𝑥

4
...
e
𝑑

𝑑𝑥 𝑓 𝑔 𝑥

= 𝑓′ 𝑔 𝑥

𝑔′ (𝑥)

5
...
e
𝑑

𝑑𝑥 𝑓 𝑥 + 𝑔 𝑥

6
...
Quotient
The derivative of the reciprocal of f (x):
𝑑

1

𝑑𝑥

𝑓 𝑥

=

𝑓 ′ (𝑥)
𝑓 (𝑥 2 )

More generally, the derivative of a quotient of functions is:𝑑 𝑔 (𝑥)
𝑓 ′ 𝑥 𝑔 𝑥 − 𝑔 ′ 𝑥 𝑓 (𝑥)
(
)=
𝑑𝑥 𝑓 (𝑥)
𝑓(𝑥 2 )
The above rules allow the derivatives of certain functions to be easily computed
...
g Rule 1 to
find the successive derivatives of:𝑓 ′ 𝑥 = 4𝑥 3 , f ′′ x = 12x 2 , f ′′′ x = f

3

x = 24x, f

4

x = 24 and f

5

x =0

Rules 1 and 5 show the first derivative of 𝑓 𝑋 = 2𝑥 2 + 4𝑥 + 1 𝑖𝑠 4𝑥 + 4, so the first
derivative is Zero when 𝑥 = −1
EXERCISE:
Find the first derivative of the functions below
(i)

𝑥 3 − 7𝑥 2 + 14𝑥 − 8

(ii)

10 − 0
...
2 FIRST ORDER DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS
The emphasis is the formation of differential equations from physical situations
...
This is referred to as the

growth constant which is a positive constant:-

𝑑𝑝
𝑑𝑡

𝛼𝑃 𝑜𝑟

𝑑𝑝
𝑑𝑡

= 𝑘𝑝

There are many physical problems which can lead to first order differential equations of
variables separable type
...
Population Growth

The population of a given species is decreased at a constant rate of n people per
annum by emigration
...
If the initial population is N people, then the
population x people after t years is given by

𝑑𝑥
𝑑𝑡

=

𝜆
100

𝑥− 𝑛

2
...
Thus,

𝑑𝑥
𝑑𝑡

= −𝜇𝑥 is a positive constant
...
Law of cooling
The rate of change of the temperature of a body is proportional to the difference
between the temperature of the body and the temperature θ of the surrounding
medium
...
Diffusion
5
...
Chemical reaction
7
...
Spread of disease

1
...
We
write it as f’’(x) or as

𝑑2 𝑓
𝑑𝑥 2


...


MAXIMUM AND MINIMUM VALUES
The value f‘ (x) is the gradient at any point but often we want to find the turning or
stationary point (maximum and minimum points) or point of inflection
...

Increasing, Decreasing or Stationary
f‘ (x) is Negative

The function is decreasing

f‘ (x) is Zero

The function is stationary (not changing)

f‘ (x) is positive

The function is increasing

Critical points include turning points and points where f‘ (x) does not exist
...
Find the first and Second order derivatives of the following:
(i)

y= 3x5

(ii)

y= 2x5 – 3x4 + 2x2 - 6
2
...
Differentiate to find the f‘(x) and f‘‘(x)
𝑓 𝑥 = 2𝑥 3 − 3 𝑥 2 − 6
𝑓 ′ 𝑥 = 6𝑥 2 − 6𝑥
𝑓 ′′ 𝑥 = 12𝑥 − 6
2
...

Put the x-value into the expression and see if it is positive, negative or zero
𝑓 ′′ 𝑥 = 12𝑥 − 6
𝑓 ′′ 0 = 12 0 − 6 = −6
𝑓 ′′ 1 = 12 1 − 6 = 6
𝑓 ′′ 𝑥 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑐𝑐𝑢𝑟𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑑𝑜𝑤𝑛
𝑓 ′′ 𝑥 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑜𝑐𝑐𝑢𝑟𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑢𝑝
𝑓 ′′ 𝑥 𝑖𝑠 𝑧𝑒𝑟𝑜, 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 (𝑚𝑎𝑦 𝑏𝑒)

4
...


𝑚𝑖𝑛𝑖𝑚𝑢𝑚

𝑚𝑎𝑦 𝑏𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛

𝑥 = 0 𝑓 ′′ 𝑥 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
′′

𝑚𝑎𝑥𝑖𝑚𝑢𝑚

𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑐𝑐𝑢𝑟𝑠

𝑥 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑜𝑐𝑐𝑢𝑟𝑠

To find the coordinates, replace the values found in step 2 into the original f (x)

𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 6
𝑓 0 = 2(0)3 − 6 = −6
𝑓 1 = 2(1)3 − 3(1)2 − 6 = −7
Maximum point (1, -7)
Minimum point (0, -6)
1
...
There are homogenous linear equations

(𝑓 𝑥 = 0), non homogenous linear equations (𝑓 𝑥 ≠ 0) and non linear equations
...
The following are examples:
1
...
Population growth of two countries
Two countries A and B have the same natural growth rate λ% per annum
...

Suppose the initial populations of A and B are N1 and N2 respectively
...

3
...
The variables X and Y
satisfy the equations

𝑑𝑥
𝑑𝑡

= −𝑥 2 𝑦,

𝑑𝑦
𝑑𝑡

= −𝑥𝑦 2
...
Hence obtain a differential equation in x and t only,
and so find an expression for x interms of t
...
Cooling of a body
5
...
Electric circuit
1
...
The
only exception is that, whenever and wherever the second variable y appears, it is
treated as a constant in every respect
...

EXAMPLE:
Suppose 𝑓 𝑥, 𝑦 = 𝑥 9 𝑦 8 + 2𝑥 + 𝑦 3
𝑓𝑥 = 9𝑥 8 𝑦 8 + 2
𝑓𝑦 = 8𝑥 9 𝑦 7 + 3𝑦 2

Then

APPLICATIONS OF DIFFERENTIATION
MAXIMUM AND MINIMUM VALUES
The concept of maximum and minimum values is termed the Zero slope analysis
...
The
concept is not only applied in plotting graphs but also to determine the maximum and
minimum values of a function such as maximizing profit or minimizing cost
...

𝑑𝑦

𝑑𝑥 = 0
𝑑𝑦

STEP 2: From the

STEP 3: Compute
If

𝑑2 𝑦
𝑑 𝑥2

This occurs at the turning point
...
If

𝑑2 𝑦

=
𝑑 𝑥2

𝑑2 𝑦
𝑑 𝑥2

> 0, the stationary

0, higher derivative (than second) can be used to

decide
...


EXAMPLE:
A company planning to have a new product in the market came up with a total sales
function S= -2000p2 + 20000p and the total cost function C= -4000p + 50000, where P and C
are the respective price and cost of the new product
...
Therefore optimum price = 6/=

APPICATION OF DIFFERENTIATION TO MARGINAL COST AND
REVENUE
It is applied to derive the relationship between Average and marginal cost and the
relationship between average and Marginal revenues
...
Therefore

𝑑

𝑇

𝑑𝑞

𝑞

= 0 i
...
Determine the maximum profit of a company with revenue function
R= 200q- 2q2 and the cost function T= 2q3 – 57q2
Profit (П) = R – T
=200q-2q2 -2q3 + 57q2
П= 200q + 55q2 – 2q3
At the turning point 𝑑П 𝑑𝑞 = 200 + 110q – 6q2= 0
100 +55q -3q2 = 0
3q2 – 55q – 100 =0
3q2 - 60q + 5q -100 =0
3q (q-20) + 5 (q -20) =0
(3q + 5) (q- 20) = 0
q = 20

0r −5 3

But d2П = 110 -12q
Dq2
: at q =20,

d2П = 110 – 240 = -130 < 0
dq2

Hence at q= 20, we have the maximum profit
...
Determine the minimum average cost if the cost function is given by:
T= 72q – 20q2 + 4q3
...


APPLICATION TO THE ELASTICITY OF DEMAND
The price elasticity of demand is the rate of change in response to the quantity
demanded to the change in price
...


𝑑𝑥

or

𝑝

{

𝑥

𝑑𝑝
𝑑𝑥

}

Where P= Price x= Quantity demanded
When Ed > 1, the demand is elastic
When Ed < 1, the demand is inelastic
...
It is
denoted by ey and defined by ey =

𝑦
𝑥


...

EXERCISE:
1
...

P= 50 – X – X2
𝑑𝑝

=

but ed =

𝑑𝑥

(

𝑝

𝑝

𝑥 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦

𝑥) (1 𝑑𝑝/𝑑𝑥)

= 50 – X- X2
X


...
0
2
...
The matrix is then referred to as an m by n matrix or (m x n) matrix
...

1 0 2

TYPES OF MATRICES
(a) Equal Matrices
Two matrices A and B are said to be equal when they have


Equal number of rows



Equal number columns



Have identical entries in the rows and columns

(b) Identity Matrix
A square matrix is said to be an identity matrix when each entry on the leading
diagonal is unity and other entries off the leading diagonal are zero
...
g
...
g
5
1
4

1 4
2 6
6 3

(d) Skew Symmetric Matrix
AT = -A
...


0
1 −4
A= −1 0
6
4 −6 0

0 −1 4
AT = 1
0 −6
−4 6
0

0 −1 4
-A = 1
0 −6
−4 6
0

(E) Null Matrix- All its entries are zero

BASIC OPERATIONS ON MATRICES
(a) Multiplication by a constant
The matrix B= CA is obtained by multiplying each entry of A by C
...
g

3 2
5 6

1
9
6
Multiplied by 3 =
4
15 18

3
12

(b) Addition of two matrices
A sum exists only when number of rows and columns of both matrices are the same
...

This is done using the same procedure as addition
...


A-B = A + (-1) B

(d ) Product of two matrices
The number of columns of A must be equal to the number of rows of B
...
2
...


2
...
2 NATIONAL INCOME MODEL
It is a simple Keynesian macroeconomic model
...



The Consumption function is

C = 𝐶0 + 𝑏 1 − 𝑡 𝑌
Where 𝐶0 , b and t are constants, b is the marginal propensity to consume
...

Rewrite the equation to become

𝑌 − 𝐶 = 𝐼0 + 𝐺0
𝑏 1 − 𝑡 𝑌 − 𝐶 = −𝐶0
OR

AV

1
−1
𝑏(1 − 𝑡) −1

𝑌
𝐶

=

=

B

𝐼0 + 𝐺0
−𝑐0

2
...
3 LEONTIF INPUT-OUTPUT MODEL
The model uses a matrix
...
There exists a production vector X which is the output
...

For each sector there is a unit consumption vector which lists the inputs needed per
unit of output of the sector
...

ASSUMPTIONS:


Each industry produces only one homogenous commodity



Each industry uses a fixed input ratio for the production of its output



Production in every industry is subject to constant returns to scale

STRUCTURE OF AN INPUT-OUTPUT MODEL WITH n INDUSTRIES

𝐴𝑛

𝑥 𝑛

=

a11
a21
an1

a12
a22
an2

a1n
a2n
ann

Where


Aij

measures how much of the i

of the

th

commodity is used for the production of each unit

jth commodity
...
g the second column states that to produce a unit of a

commodity, the inputs needed are a12 of commodity 1 and a22 of commodity 2


If no industry uses its own product as an input, the elements in the principal diagonal
of the Matrix A will all be zero
...
Such a sector
accommodates the consumer households, government sector and the foreign countries
...
+ 𝑎1𝑛 𝑥 𝑛
+ 𝑑1
𝑥2
= 𝑎21 𝑥1
+ 𝑎22 𝑥2 +
...


...


𝑥 𝑛 = 𝑎 𝑛1 𝑥1 + 𝑎 𝑛2 𝑥2 +
...

i
...
− 𝑎1𝑛 𝑥 𝑛

= 𝑑1

− 𝑎21 𝑥1 + (1 − 𝑎22 )𝑥2 −
...


...


...
+ 1 − 𝑎 𝑛𝑛 𝑥 𝑛

= 𝑑𝑛

In a matrix notation, we have

𝐼− 𝐴 𝑋= 𝑑

1 − 𝑎11
𝑎21
𝑎 𝑛1

𝑎12
1 − 𝑎22
𝑎 𝑛2

𝑎1𝑛
𝑎2𝑛
1 − 𝑎 𝑛𝑛

𝑥1
𝑥2
𝑥𝑛

=

𝑑1
𝑑2
𝑑𝑛

 I – A is called the Leontief matrix
-If it is non singular, we have
𝑥 ∗ = (1 − 𝐴)−1 𝑑
AN APPLICATION TO INPUT-OUTPUT ECONOMIC MODELS
A certain Primitive society had three industries
...
Each of these industries consumed a certain proportion of the total
output of each commodity according to the following table:
OUTPUT
FARMING

GARMENT

FARMING

0
...
2

0
...
2

0
...
4

GARMENT 0
...
2

0
...

E
...
the farming industry receives P1 for its production in any year
...
4 P1 + 0
...
3 P3 = P1
A similar analysis of the other two industries leads to the following system of equations
0
...
2 P2 + 0
...
2 P1 + 0
...
4 P3 = P2
0
...
2 P2 + 0
...
4
E=

0
...
3

0
...
6 0
...
4

0
...
3

P1
and

P=

P2
P3

The equations can be written as the homogenous system
( I – E) P = 0
Where I is the 3 x 3 identity matrix, and the solutions are
2t
P=

3t
2t

Where t is a parameter
...


EXAMPLE:
Ms
...

August was the last month for this year‘s models, and next year‘s models were introduced in
September
...
36,000

SEPTEMBER SALES

Luxury

Compact

Kshs
...
144,000 Kshs 288,000
=A

Petris

Kshs
...
0

=B
Kshs
...
216,000

(a) What were the combined dollar sales in August and September for each sales Person
and each model
(b) What was the increase in Kshs
...

3
...
Reduce cost: It is cheaper to collect data from a part of the whole population and
economical
2
...

3
...

4
...
Accuracy: The level of accuracy of a survey is assessed from the size of the sample
taken, with high quality data, there is confidence in the assessment
6
...
Much easier: It is much easier to collect information from many individuals in a
universe
8
...
This involves destroying unfit products
...
Careful sampling selection is difficult
2
...
If information for each and every unit in the study, it is difficult to interview each and
every person in sampling method

3
...
The number of units in the sample is known as sample size
...
After
inspecting the sample we draw the conclusion to accept it or reject it
...
So buy the whole quantity
only on the basis of a sample
...
It is based upon the following two conditions
...

ii)
Random selection: the sample should be selected randomly in which each and
every unit of the universe has an equal chance of being selected
...
For example if we have to study the weight of the
students studying in a college then fairly adequate sample of the students help us to arrive at
good results
...

4) Principle of optimization: – this principle states that with the help of sample one must
be able to get optimum results with maximum efficiency and minimum cost
...

SAMPLING
Once you have decided to carry out a sample survey, there are decisions which have to be
made before you start collecting information
...

NB: Bias can rarely be eliminated completely



Decide on the Sampling method

3
...
In non probability sampling, the probability that any item in the population
will be selected for the sample cannot be determined
...
Simple random sampling
This is a method of sampling in which every member of the population has an equal
probability of being selected
...
The advantage of this method is that it always produces
an unbiased sample while its disadvantage is that sampling units may be difficult or expensive
to contact
...
Systematic sampling
Sometimes called Quasi-random sampling
...
First you decide the size of the sample and then divide it into the population to
calculate the proportion of the population you require
...
Hence one of its major advantages is the speed with which it
can be selected
...


3
...
The sum of these
samples is equal to the size of the sample required and the individual sizes are
proportionate to the sizes of the strata in the population
...

4
...
Cluster sampling
Non-Probability Sampling:
1
...
This number is called a quota, which is broken down into social class, sex or
age
...
The advantage is
that it is probably the cheapest way of collecting data
...
It depends much on the
judgment and integrity of the interviewers
...
Judgmental Sampling
The researcher uses his judgment to choose appropriate members of the population for
the sample
...

3
...

4
...
4 SAMPLING AND NON SAMPLING ERROR:
The above are the two types of errors in sample surveys
Sampling Error
Samples are part of the Population and do not offer the full information about the population
...
This difference
between statistics and its estimate (parameters) due to the sampling process is called the
sampling error
...

3
...
As the sample size
increases, the sample relative frequency in any class interval gets closer to the true population
relative frequency
...

A sampling distribution shows how a statistic would vary in repeated data production
...
6 CENTRAL LIMIT THEORY
The distribution of a sample mean will tend to the normal distribution as sample size
increases, regardless of the population distribution
...
Even though the population may be non-normal
...

NORMAL DISTRIBUTION
The normal distribution is the most versatile of all the continuous probability
distributions
...
It is found useful in characterizing
uncertainties in many real-life processes, in statistical inferences, and in
approximating other probability distributions
...
7 ESTIMATION OF PARAMETERS
If data is collected from a normal distribution with mean μ and standard deviation σ
then; The sampling distribution of X, is a normal distribution with mean μx = μ and
standard deviation

σx

σ
=

√n

Summary


The mean of the sampling distribution of X is equal to the mean of the original
population: μx = μ



The standard deviation of the sampling distribution of X (also called the standard error

of the mean) is equal to the standard deviation of the original population divided by the
σ
square root of the sample size: σx =
√n


The distribution of x is (exactly) normal when the original population is normal



The Central Limit Theorem says the distribution of x is approximately normal
regardless of the shape of the original distribution, when the sample size is large
enough!

EXAMPLE:
A recent report stated that the average cost of a hotel room in Toronto is $109/ day
...

1
...
Suppose the actual sample mean cost for the sample of 50 hotel rooms is $120/Day
...
The
CLT applies
...
83

1
...
0793

105

𝑋
𝜎 / √𝑛

;

X

-1
...
83

)

= P (Z ≤ -1
...
0793
2
...
83

)

= P (Z ≥ 3
...
0000- 0
...
0001
Since the observation is so small, this suggests the observation of $120 is very rare (if
mean cost is really $109)
...


3
...
e
...
e statistics
...

Standard Errors of Common statistics:
STATISTIC
1
...
E)
σ
√n

2
...
Single Proportion (P)

√( PQ/n)

4
...
It is used when n/N is greater than 0
...




Confidence Level: The confidence level is the probability value associated with a
confidence interval
...
For example say a
confidence level of 95% (1- 0
...




Interval Estimation

3
...
It
specifies two values that contain unknown parameter
...
e
...
Then (t‘, t‘‘) is called confidence interval
...
g 5% or 1%
1-a is called confidence level e
...
10 TEST OF HYPOTHESIS
Statistical inference refers to the process of selecting and using a sample statistic to draw
conclusions about the population parameter
...
Testing of hypothesis
2
...
More precisely, it is a quantitive statement
about a population, the validity of which remains to be tested
...

TESTING HTYPOTHESIS
Testing of hypothesis is a process of examining whether the hypothesis formulated by the
researcher is valid or not
...


3
...
1 Procedure for Testing Hypothesis
1
...
These are the Null hypothesis and the alternative hypothesis
...
It states that there is no significant
difference between the sample and population regarding a particular matter under
consideration
...
When a null hypothesis is rejected, we accept the other hypothesis, which is the
alternate hypothesis
...

H1: The medicine is effective in curing cancer
2
...
E
...
The level of significance is denoted by α (Alpha)
...

3
...
t
...

4
...
Make a decision; If the calculated value is more than the table value, we reject the null
hypothesis and accept the alternative hypothesis
...



If the test statistic falls in the critical region:
Reject H0 in favour of H1



If the test statistic does not fall in the critical region:
Conclude that there is not enough evidence to reject H0
3
...
2 Type 1 and Type II errors
Type I Error: In a hypothesis test, a type I error occurs when the null hypothesis is rejected
when in fact it is true; that is, H0 is wrongly rejected
...

Type II error = (Accept H0/ H0 is not true)
3
...
3 One Tailed and Two Tailed test
One Tailed Test: Here the alternate hypothesis HA is one sided and we test whether the test
statistic falls in the critical region on only one side of the distribution

Two Tailed Tests: Here the alternate hypothesis HA is formulated to test for difference in
either direction

Common Test Statistic

One Sample Z-test

Two Sample Z-test

CRITICAL VALUES:
The critical value(s) for a hypothesis test is a threshold to which the value of the test statistic
in a sample is compared to determine whether or not the null hypothesis is rejected
...
58

1
...
33

1
...
10
...
e n ≤ 30)
...
They are defined in terms of ―number of degrees of freedom)‖
...

Degrees of freedom are number of useful items of information generated by a sample of given
size with respect to the estimation of a given population parameter or the total number of
observations minus the number of independent constraints imposed on the observations
...
g X = A +B +C

n= 4

k= 3

n –k = 1, so 1 degree of freedom
...
When the
sample size is large than 30, then sampling distribution of mean will follow Normal
distribution
...
T-table gives the
probability integral of t- distribution
...
10
...
S
...




T-test is applied when the test statistic follows t-distribution

Properties of t-distribution:


Ranges from –∞ to ∞



Bell shaped and symmetrical around mean zero



Its shape changes as the number of degrees of freedom changes
...




Variance is always greater than one and is defined only when v≥3, given as
Var (t) = ( 𝑣 𝑣 − 2)



It is more platykurtic (less peaked at the centre and higher in tails) than normal
distribution
...
As n gets larger, t-distribution
approaches normal form
...


F-TEST
F-test is used to determine whether two independent estimates of population variance
significantly differ or to establish both have come from the same population
...
It is also called Variance Ration Test
...
So it lies
between 0 and ∞



Shape of f-distribution depends upon the number of degrees of freedom



Mean = V2 / (-V2 -2), for V2 >2

EXAMPLES:
1
...
D of 120 hours
...
Using 5% level of significance, is the claim acceptable?
SOLUTION:
H0: μ = 1600
H1: μ ≠ 1600
Since the sample is large apply z-test

Z =

Difference between X and μ
SE

S
SE = √n

=

120
√100

=

120
10

= 12

1600 - 1570
Z=
12
= 30
12

= 2
...
f
...
96
...
There is significant difference between mean
life of sample and the mean life of population
...
A factory was producing electric bulbs of average length of 2000 hours
...
A sample of 25 bulbs produced by the new process were examined and the average
length of life was found to be 2200 hours
...
05)
SOLUTION:
H0: μ = 2000
H1: μ > 2000
Since sample is small, apply test
...
33

Table value of‗t‘ at 5% significance level and 24 d
...
= 1
...

; We reject the null hypothesis and accept alternative hypothesis
...
e
...
0 CHAPTER FOUR:
PROBABILITY THEORY
4
...
Probability is used in
playing games of chance such as card games, slot machines or Lotteries
...
Probability is the basis of inferential
statistics for example, predictions are based on probability and hypotheses are tested
by using probability
...
It is a number lying between 0 and 1
...

TERMS USED IN PROBABILITY
Random experiment- This is an experiment that has two or more outcomes which
vary in unpredictable manner from trial to trial when conducted under uniform
conditions
Sample Point- This is every indecomposable outcome of a random experiment
...
It is the set of all possible outcomes of an
experiment
...
g when a coin is tossed, the sample space is (Head, Tail)
Event- An event is the result of a random experiment
...
E
...

p(S) = 1
 If A and B are mutually exclusive, then:
p(A

B) = p(A) + p(B)

P roba bi li ty P roper tie s



The sum of the probabilities of an event and its complementary is 1, so the
probability of the complementary event is:



The probability of an impossible event is zero
...




If an event is a subset of another event, its probability is less than or equal
to it
...
, A k are mutually exclusive between them, then:



If the sample space S is finite and an event is S = {x 1 , x 2 ,
...
This is especially (but not exclusively) true in situations
where actions are taken or decisions are made sequentially
...

EXAMPLE
Use a Probability tree to find the sample space for the gender of children if a family
has three children Use B for boy and G for girl
...

(b) Hospital records indicated that knee replacement patients stayed in the hospital for the
number of days shown in the distribution
...



The complement rule
The Rule of Complements defines the probability of the complement of an event in terms
of the probability of the original event
...

The complement of set A, denoted by A , is a subset, which contains all outcomes, which
do not belong to A





The conditional probability rule
As a measure of uncertainty, probability depends on information
...
Thus, the probability we would give the event "Xerox stock price will go up
tomorrow" depends on what we know about the company and its performance; the
probability is conditional upon our information set
...
We may define the probability of event A conditional upon the occurrence of
event B
...

The product rule
The Product Rule (also called Multiplication Theorem) allows us to write the probability
of the simultaneous occurrence of two (or more) events
...
2 The role of Probability in Business Management
Probability distributions can be a great tool for estimating future returns and profitability
...
For
example, a company might have a probability distribution for the change in sales given a
particular marketing campaign
...

 Scenario Analysis
Probability distributions can be used to create scenario analyses
...
For example, a business might create three
scenarios: worst-case, likely and best-case
...

 Sales Forecasting
One practical use for probability distributions and scenario analysis in business is to predict
future levels of sales
...
Using a scenario
analysis based on a probability distribution can help a company frame its possible future
values in terms of a likely sales level and a worst-case and best-case scenario
...

 Risk Evaluation
In addition to predicting future sales levels, probability distribution can be a useful tool for
evaluating risk
...
If
the company needs to generate $500,000 in revenue in order to break even and their
probability distribution tells them that there is a 10 percent chance that revenues will be less
than $500,000, the company knows roughly what level of risk it is facing if it decides to
pursue that new business line
...
3 EXPECTED VALUES
(I): Discrete Random Variable:
Let a1 , a 2 , , a n ,  be all the possible values of the discrete random variable X and f (x) is
the probability distribution
...
Let  
of the discrete random variable X is

E (X )

f (x)

is

be the expected value of X
...
Then, the expected value of the continuous random
variable X is
b

E ( X )     xf ( x)dx
a

(iv)

Variance of a continuous random variable

Let the continuous random variable X taking values in [a,b] and

f (x)

is the probability

distribution
...
Then, the variance of the
continuous random variable X is
b

Var( X )    EX  E ( X )   ( x  u ) 2 f ( x)dx
2

2

a

4
...
An arrangement where order is not
important is called combination
...
The order is important for a
permutation
...

PERMUTATION
An arrangement where order is important is called a permutation
...
In how many ways can these offices be filled? 4 x 3 x 2 = 24
...


COMBINATION
An arrangement where order is not important is called combination
...
Charles has four coins in his pocket and pulls out three at one time
...
A basket contains 10 mangoes
...
In how many ways can five books be arranged on a book-shelf in the library?
II
...
Seven students line up to sharpen their pencils
...
A DJ will play three CD choices from the 5 requests
...

She decides to rank each location according to certain criteria, such as price of the store and
parking facilities
...
5 PROBABILITY DISTRIBUTION
4
...
1 NORMAL DISTRIBUTION
There will be many, many possible probability density functions over a continuous range of
values
...

If X is normally distributed with mean µ and standard deviation σ, we write

X ∼ N μ, σ2)

µ and σ are the parameters of the distribution
...
e
...
For a
continuous probability distribution we calculate the probability of being less than some value
x, i
...
P(X < x), by calculating the area under the curve to the left of x
...
5

Calculating this area is not easy and so we use probability tables
...
All we
have to do is identify the right probability in the table and use it
...
5
...
g
...
g
...
g
...
g
...
of “successes”
then the probability of observing x successes out of n trials is given by
P (X = X ) = nCxpx = nCxpx (1 – p) (n-x)

x= 0, 1 ,……
...
We say X follows a binomial
distribution with parameters n and p
...
5 the distribution will exhibit POSITIVE SKEW
 if p =0
...
5 the distribution will exhibit NEGATIVE SKEW

4
...
3 POISSON DISTRIBUTION
The Poisson distribution is a discrete probability distribution for the counts of events that
occur randomly in a given interval of time (or space)
...


Note e is a mathematical constant
...
718282
...

If the probabilities of X are distributed in this way, we write

X ∼ Po λ
λ is the parameter of the distribution
...
In contrast, the
Binomial distribution always has a finite upper limit
...

For example,

4!  4  3  2  1  24
...

FACTORIAL RULE
A collection of n different items can be arranged in order n! different ways
...
)
PERMUTATIONS RULE (WHEN ITEMS ARE ALL DIFFERENT)
Requirements:
1
...
(This rule does not apply if some of the items
are identical to others
...
We select r of the n items (without replacement)
...
(The permutation of
ABC is different from CBA and is counted separately
...
There are n items available, and some items are identical to others
...
We select all of the n items (without replacement)
...
We consider rearrangements of distinct items to be different sequences
...
nk alike, the
number of permutations (or sequences) of all items selected without replacement is

n!
n1!
...
nk!
COMBINATIONS RULE
Requirements:
1
...

2
...

3
...
(The combination of
ABC is the same as CBA
...


5
...
1 GENERALIZED LINEAR PROGRAMMING MODEL
A generalized linear model (or GLM) consists of three components:
1
...
In the initial formulation of GLMs, the distribution of
Yi was a member of an exponential family, such as the Gaussian, binomial, Poisson,
gamma, or inverse-Gaussian families of distributions
...
A linear predictor—that is a linear function of regressors, ηi = α + β1Xi1 + β2Xi2 +···+
βkXik
3
...
2 THE FUNDAMENTAL ASSUMPTIONS OF LINEAR PROGRAMMING
A problem can be realistically represented as a linear program if the following assumptions
hold:
1
...

o

This requires that the value of the objective function and the response of each
resource expressed by the constraints is proportional to the level of each
activity expressed in the variables
...
In other
words, there can be no interactions between the effects of different activities;
i
...
, the level of activity X1 should not affect the costs or benefits associated
with the level of activity X2
...
Divisibility -- the values of decision variables can be fractions
...

3
...

4
...

5
...
Linear programming problems are
distinctive in that they are clearly defined in terms of an objective function, constraints and
linearity
...
The major
characteristics include the following:


Optimization

All linear programming problems are problems of optimization
...
Thus, linear programming problems are often found in economics, business,
advertising and many other fields that value efficiency and resource conservation
...



Linearity

As the name hints, linear programming problems all have the trait of being linear
...

Linearity does not, however, mean that the functions of a linear programming problem are
only of one variable
...



Objective Function

All linear programming problems have a function called the "objective function
...
g
...
The objective function is the one that the solver of a
linear programming problem wishes to maximize or minimize
...
The objective function
is written with the capital letter "Z" in most linear programming problems
...
These constraints take the form of inequalities (e
...
, "b < 3" where b may represent
the units of books written by an author per month)
...

5
...
The
profit on each Type A shed is $ 60 and on each Type B shed is $ 84
...

Solution:
(a) Unknowns
Define
X= number of Type A sheds produced each day
Y= number of Type B sheds produced each day
(b) Constraints
Machine time: 2x + 3y ≤ 30
Craftsman time: 5x + 5y ≤ 60
And

x ≥0, y≥0

(c) Profit
P = 60x + 84y
So, in summary, the linear programming problem is
Maximize P = 60x + 84y
Subject to 2x + 3x ≤ 30
X + y ≤ 12
y ≥0
x ≥0

5
...
5
...
5
...

Z = 50x + 18y

(1)

Subject to the constraints

Step 1: Since x 0, y 0, we consider only the first quadrant of the xy - plane
Step 2: We draw straight lines for the equation
2x+ y = 100

(2)

x + y = 80
To determine two points on the straight line 2x + y = 100
Put y = 0, 2x = 100
x = 50
(50, 0) is a point on the line (2)
put x = 0 in (2), y =100
(0, 100) is the other point on the line (2)
Plotting these two points on the graph paper draw the line which represent the line 2x + y =100
...
Choose a point say (1, 0) in
R1
...
Therefore R1 is the required region for the constraint

2x + y 100
...
Find the
required region say R1', for the constraint x + y

80
...
Therefore every
point in the shaded region OABC is a feasible solution of the LPP, since this point satisfies all the
constraints including the non-negative constraints
...

(A) (a)CORNER POINT METHOD
The optimal solution to a LPP, if it exists, occurs at the corners of the feasible region
...

Step 2: Find the co-ordinates of each vertex of the feasible region
...


Step 3: At each vertex (corner point) compute the value of the objective function
...

Step 1: Draw the half planes of all the constraints
Step 2: Shade the intersection of all the half planes which is the feasible region
...
Sometimes it is convenient to take k as the LCM of a and b
...
This line
should contain at least one point of the feasible region
...

To minimise Z draw a line parallel to ax + by = k and nearest to the origin
...
Find the co-ordinates of this point by solving
the equation of the line on which it lies
...
5
...
He manufactures two types of boxes using a
combination of three types of wood, maple, walnut and cherry
...
It is the volume of a one-foot length of a board one foot wide and one inch
thick) maple and 1 bf walnut
...
Given that he has 10 bf of maple, 5 bf of walnut and 11 bf of cherry and he can
sell Type I of box for $120 and Type II box for $160, how many of each box type should he
make to maximize his revenue? Assume that the woodworker can build the boxes in any size,
therefore fractional solutions are acceptable
...


Max Z  120 x1  160 x2

2 x1  10
3 x 2  11
x1  x 2  5
x1 , x 2  0
To convert the first constraint form an inequality to equality, we introduce the first slack
variable s1 where

s1  10  2 x1 or 2 x1  s1  10
...
t
...

Previously we have shown that the solution where x1  0 and x2  0 is a basic feasible solution
so we will start the algorithm here
...

At this step we create the tableau for this basic feasible solution which was initially shown in
Table 1
...

The current solution is not optimal
...
Since x 2 has the
most negative coefficient in row 0 and s 2 has the lowest ratio, the entering and the leaving
variables are x2 and s 2 , respectively
...
Return to Step 3
The new basic feasible solution is shown in Table 4, which is the same as Table 2
...
The tableau for the new basic feasible solution in the first iteration
Basic
RHS
s3
x1
s1
x2
s2
Z

Z

1

-120

0

0

160

s1
x2

0
0

2
0

0
1

1
0

s3

0

1

0

0

0
1
3
1
3

3

0
0
0
1

Ratio

1760

3
10
11
3
4
3

5
None
4

3

Step 6:
Determine if the basic feasible solution is optimal
...

Step 7:
If the current basic feasible solution is not optimal, select a non basic variable that should
become a basic variable and basic variable which should become a non basic variable to
determine a new basic feasible solution with an improved objective function value
...

Step 8:
Use elementary row operations to solve for the new basic feasible solution
...


Table 5
...

Basic

Z

x1

x2

s1

s2

s3

Z

1

0

0

0

40

s1

0

0

0

1

2

x2

0

0

1

0

1

x1

0

1

0

0

1

3
3

2240
22

0

3

120
-2

3

RHS

11

1

4

Ratio

3

3

3

3

Step 9:
Determine if the basic feasible solution is optimal
...
This solution also
corresponds to the extreme point B in Figure 4 which was also determined to be optimal using
the graphical solution approach
...
67
...
5
...

If the Z row value for one or more non basic variable is zero in the optimal tableau, alternate
optimal solution exists
When determining the leaving variable of any tableau, if there is no positive minimum ratio or
all entries of pivot column are negative or zero there is unbounded solution
If at least one artificial variable is positive in the optimum iteration, then the LPP has no
feasible solution

5
...
5 DUALITY THEOREM
It states that if the primal problem (either in symmetric or asymmetric form) has an optimal
solution, then so does the dual and the optimal values of their respective objective functions
are equal
...

 If one has no feasible solution, then
-

Either has no feasible solution

-

Or has unbounded objective function

CHAPTER SIX

6
...



A Stochastic process is a collection of random variables, representing the evolution
of some system of random values over time

6
...
This property is called Markov property
...
3 Markov chain is a discrete-time stochastic process with the Markov property
...
The “time” can be discrete i
...

the integers), continuous (i
...
the real numbers), or, more generally, a totally ordered
set
...
The outcome of each experiment is one of a set of discrete states
2
...

For example in a transition matrix, a person may be assumed to be in one of the
three discrete states (lower, middle or upper income), with each offspring in one of
these same three discrete states
...

A mouse lives in the cage
...
When the mouse is in cell 1 at time n (minutes) then, at time n + 1 it is either still in 1
or has moved to 2
...
05; it does so, regardless of where it was at earlier times
...
99
...
2TRANSITION PROBABILITIES
We can summarise the information above by the transition diagram:
α
1-β

1
1-α

2
β

Another way to summarise the information is by the 2×2 transition probability matrix
P = 1–α
β

α
0
...
05
1 − β = 0
...
01

Questions of interest:
1
...
How often is the mouse in room 1?
Question 1 has an easy, intuitive, answer: Since the mouse really tosses a coin to decide
whether to move or stay in a cell, the first time that the mouse will move from 1 to 2 will have
mean 1/α = 1/0
...
(This is the mean of the binomial distribution with parameter
α
...
4 STEADY STATE
We say that the process (Xn, n = 0, 1,
...
, for any m ∈ Z+
...
Clearly, a sequence of i
...
d
...

NB:
*m ∈ Z+ = m is an element of Z+
* i
...
d = Independent and identically distributed random variables
...
5 APPLICATIONS OF MARKOV MODELS
Speech recognition,
Modeling of coding/noncoding regions in DNA,
Protein binding sites in DNA,
Protein folding,
Protein superfamilies,
Multiple sequence alignment,
Flood predictions,
Ion channel recordings,
Optical character recognition
...
0 REGRESSION ANALYSIS
Regression is the measure of the average relationship between two or more variables in terms
of the original units of the data
...
Regression equations through regression coefficient
2
...

The two regression equations
for X on Y; X = a + bY
And for Y on X; Y = a + bX
Where X, Y are variables, and a, b are constants whose
Values are to be determined
For the equation, X = a + bY
The normal equations are
X = na + b Y and
XY = aY + bY2
For the equation, Y= a + bX, the normal equations are
Y = na + b X and
XY = aX + bX2
From these normal equations the values of a and b can be
determined
...
(2)
Multiplying (1) by 6
240 = 30a + 180b……
...
65

Now, substituting the value of ‗b‘ in equation (1)
40 = 5a – 19
...
5
a = 59
...
9
Hence, required regression line Y on X is Y = 11
...
65 X
...
(3)
214 = 40a + 340b …
...
(5)
(4) – (5) gives
-26 = 20b
b = - 26
20
= - 1
...
3 in equation (3) gives
30 = 5a – 52
5a = 82
a = 82
5
= 16
...
4 – 1
...
1 CORRELATION ANALYSIS
Correlation refers to the relationship of two variables or more
...

COMPUTATION OF CORRELATION
When there exists some relationship between two variables, we have to measure the degree of
relationship
...


KARL PEARSON’ S COEFFICIENT OF CORRELATION:
Karl Pearson, a great biometrician and statistician, suggested a mathematical method for
measuring the magnitude of linear relationship between the two variables
...
It is denoted
by ‗ r‘
...
D of x and y
(ii) r=  xy
n σx σy
(iii) r =  xy
√ x2
...

Simple formula is the third one
...

Steps:
1
...

2
...

X = x- x , Y = y- y
3
...

4
...
This is covariance
...
Substitute the values in the formula
...
√(y- y)2
n
n
The above formula is simplified as follows
r=  xy
√ x2
...
Correlation coefficient lies between –1 and +1
2
...

3
...

4
...

5
...

6
...

rxy = ryx

EXAMPLE
Find Karl Pearson‘s coefficient of correlation from the following data between height of
father (x) and son (y)
...


RANK CORRELATION COEFFICIENT
It is studied when no assumption about the parameters of the population is made
...
It is useful to study the qualitative measure of attributes like honesty,
colour, beauty, intelligence, character, morality etc
...
This method was developed by
Edward Spearman in 1904
...

 D2 = sum of squares of differences between the pairs of ranks
...


The value of r lies between -1 and +1
...
If r = -1, then there is complete disagreement in order
ranks and they are in opposite directions
...
0714 = 0
...
9286 shows that the data between X and Y is highly positively correlated

THANKS FOR YOUR TIME!



---