The Summation Notation by Q719y1

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```									                The Summation Notation
• The sum of a large number of terms occurs
frequently in econometrics. There is an
abbreviated notation for such sums. The upper
case Greek letter  (sigma) is used to indicate a
summation and the terms are generally indexed
by subscripts.
4
The symbol  X i read ``the sum of X sub i as i goes from
i1

1 to 4,'' represents the sum X1 + X 2 + X 3 + X 4 .
Sisira Sarma           18.317: Introduction to Quantitative Methods in Economics
Examples
5
Example 1:   X
i 1
i        X1 + X 2 + X 3  X 4  X 5
7
Example 2:    (2k -1) =49
k=1
3
Example 3:    ( X i  3) 2  ( X 1  3) 2  ( X 2  3) 2  ( X 3  3) 2
i 1
2
             
Example 4:   ( X i  3)   X 1  X 2  X 3  X 4  X 5  15
5
2

 i 1        
2
Example 5:    (X
i 1
i    3)(Yi  5)  ( X 1  3)(Y1  5)  ( X 2  3)(Y2  5)

Sisira Sarma                       18.317: Introduction to Quantitative Methods in Economics
Example
• If five people are asked their ages the
results might be summarized in the table
Person          1   2               3                  4                 5
Age (X) 23          19              40                 22                35

and the sum of their ages can be represented
as 23 + 19 + 40 + 22 + 36 = 140

Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Properties of the Summation Operator
n
Property 1: If k is a constant  k  nk
i 1
n                  n
Property 2:      kX
i 1
i    k  Xi
i 1

 X        Yi  
n                         n                     n
Property 3:
i 1
i                   X Y
i 1
i
i 1
i

 a  bX   na  b X ; a and b are constants
n                                         n
Property 4:                    i                                      i
i 1                                      i 1

Sisira Sarma               18.317: Introduction to Quantitative Methods in Economics
Sample Average

Sample Average (or sample mean):
n
1
X   Xi
n i 1
n
       X
i 1
i    n X
Sisira Sarma          18.317: Introduction to Quantitative Methods in Economics
Results
n
1. If di  X i  X then show that  di  0
i=1

 X               X         X                       n X 
n                             n
2                              2                   2
2.                  i                               i
i=1                            i=1

 X               X Yi  Y    X i Yi  n X  Y 
n                                              n
3.                  i
i=1                                          i=1

Sisira Sarma             18.317: Introduction to Quantitative Methods in Economics
Example

Person         1    2                 3                 4                  5
Age (X)        23   19                40                22                 35

• Sample mean = (23 + 19 + 40 + 22 + 36)/5
= 140/5 = 28
• And, (23 – 28 + 19 – 28 + 40 – 28 + 22 – 28
+ 36 –28) = 140 – 140 = 0.
• Verify the results 2 and 3 as well.
Sisira Sarma        18.317: Introduction to Quantitative Methods in Economics
Some Concepts
• Variable: Any measurable characteristic of a set of data
is called a variable.
• Discrete Variable: A variable is said to be discrete if it
can assume only a finite or countably infinite
number of values.
• Continuous Variable: A variable is said to be
continuous if it can assume any values whatsoever
between certain limits. Examples of continuous
variables include those representing length, weight,
and time.
Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• Random Experiment: A random experiment is a
process leading to at least two possible
outcomes with uncertainty as to which will occur.
Examples: Tossing a coin, throwing a pair of
dice, drawing a card from a pack of cards are all
experiments.
• Sample Space: The set of all possible outcomes of an
experiment is called the sample space (or
population).
Sisira Sarma     18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• Exercise: Define Sample Space for a) tossing two fair
coins, b) tossing three fair coins and c) throwing a
pair of dice.
• Sample Point: Each member, or outcome of the sample
space (or population) is called a sample point.
• Event: An event is a subset of the set of possible
outcomes of an experiment. Example: Let event A
be the occurrence of 'one head and one tail' in the
experiment of tossing two fair coins. You can see
that only outcomes { HT,TH} belong to event A.
Sisira Sarma      18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• Mutually exclusive events: Events are said to
be mutually exclusive if the occurrence of
one event prevents the occurrence of another
event at the same time.
• Equally likely events: Two events are said to be
equally likely if we are confident that one
event is as likely to occur as the other event.
Example: In a single toss of a coin a head
is as likely to appear as a tail.
Sisira Sarma    18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• Collectively exhaustive: Events are said to be collectively
exhaustive if they exhaust all possible outcomes of an
experiment. Example: In our two coin tossing
experiment {HH,HT,TH,TT} are the possible
outcomes, they are (collectively) exhaustive events.
• Random Variable (r.v.): A variable that stands for the
outcome of a random experiment is called a random
variable. A random variable satisfies four properties:
– 1) it takes a single, specific value; 2) we do not know in
advance what value it happens to take; 3) we do, however
know all of the possible values it may take; and 4) we know
the probability that it will take any one of those possible
values.
Sisira Sarma        18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• Examples of r.v: The result of rolling of a die,
tomorrow's stock price, tomorrow's exchange
rate, GNP, money supply, wages, etc.
• Discrete r.v.: A random variable that takes a
finite number of values is called a discrete
random variable.
• Continuous r.v.: A random variable that can
take any value within a range of values is called
a continuous random variable.
Sisira Sarma    18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• The probability of an event (classical
approach): If an experiment can result in n
mutually exclusive and equally likely outcomes
and m of these outcomes are favourable to
event A, then the probability that A occurs (
denoted as P(A)) is the ratio m/n.

m the number of outcomes in the event
P A  
n the total number of possible outcomes
Sisira Sarma    18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• What happens if the outcomes of an experiment are not
finite or not equally likely?
• The probability of an event (relative frequency or
empirical approach): The proportion of time that an
event takes place is called its relative frequency, and
the relative frequency with which it takes place “in the
long run” is called its probability. If in n trials, m of
them are favourable to event A , then P(A) = m/n,
provided the number of trials are sufficiently large
(technically, infinite).                       m
P A  lim                        .
n       n
Sisira Sarma      18.317: Introduction to Quantitative Methods in Economics
Probability and Statistics: Basics
• There is yet another definition of probability, called as
the subjective probability, which is the foundation of
Bayesian Econometrics. Under the subjective or
“degrees of belief”definition of probability, you can ask
questions such as:
• What is the probability that Iraq will have a democratic
government?
• What is the probability that terrorists will attack the
United States in November 2004 (the presidential
election time)?
• What is the probability that there will be a stock market
boom in 2005?
Sisira Sarma      18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
1. A probability cannot be  negative or larger than 1.
That is, 0  P A  1 for any event A.
2. If the probability of the occurrence of event A is P(A),
the probability that A will not occur is 1-P(A).
3. Special rule of addition: If A and B are mutually exclusive events,
the probability that one of them will occur is P(A or B) = P( A) +P( B).
Similarly, if A1,A2 ,...,An are mutually exclusive events,
the probability that one of them will occur is P(A1 or A2 or ... or An ) =
P( A1  A2  ... An )  P( A1 ) +P( A2 ) ... P( An ).

Sisira Sarma             18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
4. If A1,A2 ,...,An are mutually exclusive and collectively
exhaustive set of events, the sum of the probabilities of
their individual occurences is 1. That is,
P( A1  A2  ... An )  P( A1 )  P( A2 ) ... P( An )  1.

Example:The probability of any of the six numbers on a die is
1/6 since there are six equally likely outcomes and each one of
them has an equal chance of turning up. Since the numbers
{1,2,3,4,5,6} form an exhaustive set of events
P(1+2+3+4+5+6) = P(1)+P(2)+P(3)+P(4)+P(5)+P(6) = 1.
Sisira Sarma          18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
• Independent Events: Two or more events are said to
be independent if the occurrence or non-occurrence of
one does in no way affect the occurrence of any of the
others. Note: Mutually exclusive events are necessarily
independent. However the converse is not necessarily
the case.
• 5. (Special rule of multiplication) If A and B are
independent events, the probability that both of them
will occur simultaneously is P(AB) = P(A and B) =
P(A).P(B). Since P(A and B) means the probability of
events A and B occurring simultaneously or jointly, it is
called a joint probability.
Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
Example: Suppose we flip two identical coins
simultaneously. What is the probability of
obtaining a head on the first coin (call event A)
and a head on the second coin (call event B)?
Notice that probability of obtaining a head on the
first coin is independent of the probability of
obtaining a head on the second coin. Hence,
P(AB) = P(A).P(B) = (½).(½) = ¼.

Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
• 3b (modification to rule 3): If events A and B are not
mutually exclusive, then P(A+B) = P(A) + P(B) –
P(AB).
• Example: A card is drawn from a well shuffled pack of
playing cards. What is the probability that it will either
• Notice that spade and queen are not mutually exclusive
events – one of the 4 queens is spade! So, P(a spade or
13/52+4/52 – 1/52 = 16/52 = 4/13.
Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
• Conditional Probability: The probability that event B
will take place provided that event A has taken place
(is taking place or will with certainty take place) is
called the conditional probability B relative to A.
Symbolically, it is written as P(B|A) to be read “the
probability of B, given A.”
• If A and B are mutually exclusive events, then P(B|A)
= 0 and P(A|B) = 0.
• 6. (General rule of multiplication) If A and B are any
two events, then the probability of their occurring
simultaneously is P(A and B) = P(A).P(B|A) =
P(B).P(A|B).
Sisira Sarma       18.317: Introduction to Quantitative Methods in Economics
Rules of Probability
• This implies that P(B|A) = P(AB)/P(A) and P(A|B) =
P(AB)/P(B).
• Example: In a Principles of Economics class there are 500 students
of which 300 students are males and 200 are females. Of these, 100
males and 60 females plan to major in economics. A student is
selected at random from this class and it is found that this student
plans to be an economics major. What is the probability that the
student is a male?
• Define A be the event that the student is a male and B be the event
that the student is an economics major. Thus, we want to find out
P(A|B).
• P(A|B) = P(AB)/P(B) = (100/500)/(160/500) = 0.625. (conditional)
• What is P(A)? (300/500) = 0.6 (unconditional)
Sisira Sarma         18.317: Introduction to Quantitative Methods in Economics
Exercise
1. The Experiment: Flipping a fair coin three times in a row.
• 1a. Let A be the event of getting exactly two heads. What is P(A)?
• 1b. Let B be the event of getting a tail on the first flip. What is
P(B)?
• 1c. Let C be the event of getting no tails. What is P(C)?
2. For the three-flip activity described above, are the two events in
each of the following pairs mutually exclusive?
• 2a. A and C.
• 2b. B and C.
• 2c. A and B.
• 2d. Two tails, two heads.
• 2e. Head on first flip, two tails.
• 2f. Tail on first flip, tail on third flip.
Sisira Sarma         18.317: Introduction to Quantitative Methods in Economics
Outcomes: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT};
denote these outcomes as E1, E2, E3, E4, E5, E6, E7 and E8,
respectively.
Notice that the probability of each event is 1/8.
1a. Let A be the event of getting exactly two heads. What is P(A)?
• P(A) = P(E2)+P(E3)+P(E4) = 1/8 + 1/8 + 1/8 = 3/8.
1b. Let B be the event of getting a tail on the first flip. What is P(B)?
• P(B) = P(E4) + P(E6) + P(E7) + P(E8) = 1/8+1/8+1/8+1/8 =
½.
1c. Let C be the event of getting no tails. What is P(C)?
• P(C) = P(E1) = 1/8.

Sisira Sarma          18.317: Introduction to Quantitative Methods in Economics
Outcomes: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT};
denote these outcomes as E1, E2, E3, E4, E5, E6, E7 and E8,
respectively.
Let A be the event of getting exactly two heads: A = {E2, E3, E4}
Let B be the event of getting a tail on the first flip: B = {E4, E6, E7,
E8)
Let C be the event of getting no tails: C = {E1}.
Let D be the event of two tails: D = {E5, E6, E7}.
Let E be the event of getting head on first flip: E = {E1, E2, E3, E5}.
Let F be the event of getting tail on third flip: F = {E2, E5, E6, E8}.

Sisira Sarma          18.317: Introduction to Quantitative Methods in Economics
A = {E2, E3, E4}                  B = {E4, E6, E7, E8)
C = {E1}                          D = {E5, E6, E7}.
E = {E1, E2, E3, E5}              F = {E2, E5, E6, E8}
• 2a. A and C – mutually exclusive.
• 2b. B and C - mutually exclusive.
• 2c. A and B – not mutually exclusive because E4 is common.
• 2d. Two tails, two heads: A and D – mutually exclusive.
• 2e. Head on first flip, two tails: E and D – not mutually exclusive
because E5 is common..
• 2f. Tail on first flip, tail on third flip: B and F – not mutually
exclusive because E6 and E8 are common.

Sisira Sarma         18.317: Introduction to Quantitative Methods in Economics
Exercise (contd.)
A = {E2, E3, E4},     B = {E4, E6, E7, E8) and C = {E1}.
1. What is P(A and B)
2. What is P(A and C)
3. What is P(B and C)
4. What is P(A or B)?
Answer: P(A) + P(B) – P(AB) = 3/8 + 4/8 – 1/8 = 6/8 = ¾.
5. What is P(A|B)?
Answer: P(AB)/P(B) = (1/8)/(4/8) = ¼.
Sisira Sarma         18.317: Introduction to Quantitative Methods in Economics
Exercise (contd.)
Let B be the event of getting a tail in the first flip.
Let G be the event of getting a tail in the second flip.
B = {E4, E6, E7, E8)
G = {E3, E5, E7, E8}
Question: Are B and G independent?
Recall the definition of independent: If A and B are independent
events, the probability that both of them will occur
simultaneously is P(AB) = P(A and B) = P(A).P(B). That is, A
and B are said to be independent, P(A|B) = P(A).