The Summation Notation

1
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.

2
Examples
3
Example

• If five people are asked their ages the results
  might be summarized in the table
• and the sum of their ages can be represented as
  23 19 40 22 36 140

4
Properties of the Summation Operator
5
Sample Average
6
Results
7
Example

• 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.

8
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.

9
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).

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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.

11
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.

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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.

13
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.

14
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.

15
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).

16
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 beliefdefinition
  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?

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Rules of Probability
18
Rules of Probability
ExampleThe 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(123456) P(1)P(2)P(3)P(4)P(5)P(6)
1.
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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.

20
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.

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Rules of Probability

• 3b (modification to rule 3) If events A and B
  are not mutually exclusive, then P(AB) 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 a spade or a queen?
• Notice that spade and queen are not mutually
  exclusive events one of the 4 queens is spade!
  So, P(a spade or a queen) P(spade)P(queen)
  P(spade and queen) 13/524/52 1/52 16/52
  4/13.

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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(BA) to be read the probability of
  B, given A.
• If A and B are mutually exclusive events, then
  P(BA) 0 and P(AB) 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(BA) P(B).P(AB).

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Rules of Probability

• This implies that P(BA) P(AB)/P(A) and P(AB)
  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(AB).
• P(AB) P(AB)/P(B) (100/500)/(160/500)
  0.625. (conditional)
• What is P(A)? (300/500) 0.6 (unconditional)

24
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.

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Answers

• 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)
  1/81/81/81/8 ½.
• 1c. Let C be the event of getting no tails. What
  is P(C)?
• P(C) P(E1) 1/8.

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Answers (contd.)

• 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.

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Answers (contd.)

• 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.

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Exercise (contd.)

• A E2, E3, E4, B E4, E6, E7, E8) and C
  E1.
• 1. What is P(A and B)
• Answer P(E4) 1/8.
• 2. What is P(A and C)
• Answer ?
• 3. What is P(B and C)
• Answer ?
• 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(AB)?
• Answer P(AB)/P(B) (1/8)/(4/8) ¼.

29
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(AB) P(A).
• Answer
• P(B) 4/8 ½ P(G) 4/8 ½ P(BG) 2/8
• P(GB) P(GB)/P(B) (2/8)/(4/8) ½.