Quantitative Analysis

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Quantitative Analysis

• BEO6501

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Themes

• These lectures will deal with the measurement of
  uncertainty or chance.
• In particular
• Definition and meaning of probability.
• Laws of probability.
• Expected values.
• Probability distributions and random numbers.

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Reference

• Refer to Lapin
• Probability Concepts.
• Probability Distributions, Expected Value, and
  Sampling.
• Almost any intermediate business statistics text
  will have equivalent chapters or sections that
  would be useful.

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Definition

• The problem
• Selvanathan, for example, notes that definitions
  are often circular.
• Berenson Levine use The likelihood or chance
  that a particular event will occur.
• It feels like probability, chance and
  likelihood are merely synonyms for one another.
• Probability is all about measuring likelihood or
  whatever we want to call it.
• I.e. Attaching numbers to the chance that a
  particular event will happen.
• Numbers that tell us what is more likely or less
  likely.

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Approaches to probability

• Relative frequency
• Think of probability as the relative frequency of
  a particular event occurring under the same
  circumstances.
• E.g. Prhead 0.5 means that if we spin a coin
  many times, we expect the result heads to occur
  in 50 percent of the trials.
• Same circumstances?
• Lots of spinning.
• Unweighted coin.
• An upshot.
• This definition implies that probability
  statements regarding sporting events are
  meaningless because the circumstances are never
  the same from one even to the next.

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Approaches (cont.)

• A priori or classical probability
• We use logic and/or mathematics to determine
  probabilities.
• E.g. Probabilities in games of chance.
• Roulette.
• Poker.
• Illustration
• We have seen 10 card dealt from a standard deck
  of 52 and have noted that one ace is amongst the
  10.
• What is the probability that the next card turned
  up will be another ace?
• We would argue as follows
• The deck has been well shuffled so any of the
  remaining cards is equally likely to be the next
  card turned up.
• There are 42 cards still in the deck.
• Any particular card has 1 chance in 42 of being
  the next card.
• There are 3 aces in the deck.
• The probability that the next card is an ace is
  3/42 or 1/14.
• It means that in circumstances described in this
  problem, we would expect to turn up an ace one
  time in 14 on average.

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Approaches (cont.)

• Statistical probability.
• We use data to determine relative frequencies and
  probabilities.
• E.g. Probabilities of industrial accidents
  occurring in particular types of workplaces.
• E.g. Probabilities of motor accidents occurring
  Victorian roads.
• We cannot figure it out from theory.
• The problem is how do we know that the same
  circumstances apply in all cases being
  considered?
• We usually dont unless we are using a controlled
  experiment.
• E.g. Using guinea pigs to test pharmaceutical
  products might come close.

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Approaches (cont.)

• More statistical probability.
• An illustration
• Say an insurance company wishes to determine the
  probability that an applicant for a motor policy
  will be involved in a motor accident.
• It would need to estimate the number of policy
  holders (or better still, drivers) in the recent
  past similar to the applicant.
• Age?
• Sex?
• Location?
• History of accidents?
• History of driving infringements?
• It would need to estimate the number of these
  involved in particular types of accidents.
• Minor?
• Serious?
• Write-off?

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Approaches (cont.)

• More statistical probability.
• An illustration
• Say 15 of male drivers who were under 25 and
  resided in a metropolitan area and had no recent
  accidents or driving infringements had a minor
  road accident in the last year.
• This would be a reasonable estimate of the
  probability that a similar applicant for
  insurance will have a minor accident in the year
  to which a policy would apply.
• Who cares?
• If the insurance company underestimates the
  probability, it may under-quote and this could
  cause its pay-outs to policy holders to exceed
  its planned budget for pay-outs (with obvious
  consequences).
• What about same circumstances?
• A wetter than usual winter?
• Safer roads?
• Safety awareness campaigns?
• Whatever the problems, insurance companies must
  estimate probabilities.
• Their viability depends on getting it close to
  right.

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Approaches (cont.)

• Subjective probabilities.
• These are not measurable, either statistically or
  theoretically, and are merely statements of
  informed or uninformed opinion.
• They are of no use in statistical (or any other)
  analysis.

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Be wary

• Recall the definition.
• Relative frequency
• Think of probability as the relative frequency of
  a particular event occurring under the same
  circumstances.
• Same circumstances seem unlikely in many
  apparently useful applications.
• All interesting questions are about future out
  comes, however
• Incomes will be different.
• Prices will be different.
• Technology will be different.
• Tastes may be different.

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Laws of probability

• Extreme cases
• For some event A
• PrA 0 means that A never happens.
• E.g. A sum of faces when a pair of dice are
  rolled.
• PrA 13 0.
• PrA 1 means that A always happens.
• E.g. A sum of faces when a pair of dice are
  rolled.
• Pr0 lt A lt 13 1.

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Laws (cont.)

• Introduction.
• It is useful to introduce the laws with an
  example using a contingency table (I.e.
  cross-tabulation).
• Say have two questions from a sample survey
• Age group.
• Response to the proposition that gambling
  facilities should not be permitted to be open 24
  hours per day.
• See the next slide for a hypothetical contingency
  table.

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Laws (cont.)

• Marginal probabilities.
• Probabilities of single events
• Suppose one of the 200 respondents is drawn at
  random.
• Prany particular questionnaire is selected
  1/200.
• At random means that each unit or element has
  an equal chance of selection.
• This means that, on average, a particular unit
  should be drawn 1 time in every 200.
• The probability is its relative frequency.
• Remembering that we are drawing one respondent at
  random, check out the next slide.

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Prperson is under 25 ?
Prperson is under 25 60/200 0.30 or 30
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Prperson agrees ?
Prperson agrees 80/200 0.40 or 40
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Laws (cont.)

• Joint probability.
• Compound events
• Terms
• and means both events happen.
• PrA and B relative frequency of both
  occurring together.
• or means at least one of the events occur.
• PrA or B PrA PrB PrA and B.
• This is the addition rule.
• Note A and B is in event A and in event B so we
  must be careful not to count it twice!
• Conditional probability
• Something about the question is known with
  certainty.
• Say B has happened or will certainly happen.
• PrA given B PrA/B PrA and B PrB.
• Check out the next slides that illustrate these
  laws.

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Prperson disagrees AND is over 55 ?
Prperson disagrees AND is over 55 5/200
0.025 or 2.5
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Prperson is neutral OR is over 55 ?
Prperson is neutral OR is over 55 95/200
47.50 or 47.5
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Prperson is neutral OR is over 55 using the
formula
Pr 45/200 60/200 10/200 95/200 47.50 or
47.5
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Prperson agrees GIVEN the person is over 55 ?
Pragrees GIVEN the person is over 55 45/60
0.75 or 75
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Pr (45/200) (60/200) 45/60 0.75 or 75
Pragrees GIVEN the person is over 55 using
the formula?
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Laws (cont.)

• Mutually exclusive events.
• For two events A and B
• If PrA and B 0 then A and B are mutually
  exclusive.
• I.e. They never occur together.
• Example
• Pr person is under 25 and over 55 0.
• It doesnt make sense.
• It cant happen.

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Laws (cont.)

• Complementary events.
• This means not A.
• I.e. Everything except A.
• Write it A and say it A complement.
• PrA PrA 1.
• A or A must happen.
• They cant both happen simultaneously.
• PrA 1 - PrA.
• PrA 1 - PrA.
• Sometimes it may be difficult to calculate a
  particular probability directly, but it might be
  easy to calculate its complement.
• If so, the formula makes it easy.

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Laws (cont.)

• Independent events.
• If A and B are independent, knowing B yields no
  information about A.
• Example
• A person weighs more than 80 kilos.
• B person is over 2 metres tall.
• A and B are almost certainly not independent
  because very tall people weigh more on average
  than others.
• Example
• A person weighs more than 80 kilos.
• B person has blue eyes.
• A and B are almost certainly independent because
  blue eyed people seem unlikely to weigh more on
  average than others (or less).
• Formulas
• PrAB PrA.
• Knowing B tells us nothing about A.
• PrA and B PrA PrB.
• The multiplication rule but only for
  independent events.

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Probability distributions

• Concept.
• A probability distribution shows
• All possible outcomes from a process.
• The probabilities of all possible outcomes.
• Types
• Discrete.
• We can list all of the outcomes.
• E.g. If we roll a pair of dice, the possible
  totals are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
• Continuous.
• We can only describe the outcomes as intervals
  (or ranges).
• E.g. If the variable in question was body
  weights, we could not list all of the
  possibilities as exact numbers.
• If we say a person is 90 kilos we probably the
  weight is between 89.5 kilos and 90.5 kilos.

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Distributions (cont.)

• Discrete.
• Poisson distribution (but there are others too).
• PrX the probability of exactly X successes
  happening.
• ? expected or average number of successes.
• e Eulers number (approximately 2.718).
• X! X factorial or X (x 1) (X 2) 3
  2 1.
• 1! 1 and 0! 1.

Although the probabilities are easy to calculate
with an electronic calculator, they are
tabulated in many texts.
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Distributions (cont.)

• Circumstances.
• The Poisson distribution is often appropriate
  where there are many events and where the
  probability of success for any individual
  event is small.
• Example.
• An airline has a fleet of Boeing 747-400 planes,
  exch with 445 passenger seats. The airline knows
  from experience that 2 percent of bookings are
  cancelled at the last minute and reselling these
  cancelled tickets is difficult. Empty seats, of
  course, represent lost revenue.
• Suppose the airline overbooks up to 450
  passengers per flight.
• What is the probability that on a fully
  overbooked flight, that all passengers who arrive
  for the flight can fly out.

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Distributions (cont.)

• Solution.
• ? expected number of cancellations.
• ? 450 0.02 9.0.
• I.e. Number of bookings probability a booking
  is cancelled.
• There is a problems if there are less than 5
  cancellations.
• We have 450 bookings and 445 seats.
• X number of cancellations.
• Check the Poisson table with ? 9.0.
• PrX 0 0.0001.
• PrX 1 0.0011.
• PrX 2 0.0050.
• PrX 3 0.0150.
• PrX 4 0.0337.
• Adding these probabilities yields 0.0549 which is
  the probability that at least one passenger on a
  fully booked flight will not find a seat.

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Continuous

• Normal distribution
• Bell shaped density function.
• Parameters
• Mean ?.
• Standard deviation ?.

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Density function

• The area under the normal curve shows the
  probability that a normal random variable will
  lie in a particular range.
• The curve is
• f(X) (2?)-½ ?-1 exp(-½(X - ?)/?2).
• It looks ferocious!
• Fortunately the results are shown in Lapin Table
  B.

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Normal curve
X
?
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? small
? large
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PrX1 ? X ? X2
X
?
X1
X2
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Standard normal

• Standard normal.
• All normal distributions (say of X) can be
  transformed to the standard normal distribution
  (z)
• Mean ? 0.
• Standard deviation ? 1.
• z (X - ?)/?.

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Example

• Data
• ? 100, ? 10 and X is normally distributed.
• Question
• PrX ? 120.
• Standard normal z
• X 120 ? z (120 - 100)/10 2.0.
• X 100 ? z (100 - 100)/10 0.0.
• Look up z 2.00 in the normal table in Lapin.
• Check it gives a probability value of 0.4772.

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From standard normal tables Pr0 ? z ? 2
0.4772.
0
2
z
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PrX ? 120 Prz ? 2 0.5000 - 0.4772
0.0228.
0.4772
0.5000
0
2
z
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Another example

• Problem
• The weekly demand at supermarket chain for 375
  gram jars of XYZ freeze dried instant coffee is
  approximately normal with mean 77.8 cases and
  standard deviation 12.9 cases.
• What is the probability that
• Demand will be between 70 and 90 cases in a
  particular week?
• Demand will be more than 100 cases in a
  particular week?
• Demand will be more than 60 cases in a particular
  week?

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Example (cont.)

• Calculate z scores.
• w 70.
• ? z (70 77.8)/12.9 - 0.60.
• w 90.
• ? z (90 77.8)/12.9 0.95.
• w 100.
• ? z (100 77.8)/12.9 1.72.
• w 60.
• ? z (60 77.8)/12.9 - 1.38.

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0
z
Pr70 ? demand ? 90 0.3289 0.2257 0.5546.
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z
0
Pr100 ? demand 0.5000 - 0.4573 0.0427.
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0
z
Pr60 ? demand 0.4162 0.5000 0.9162.
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Expected values

• Expected value mean (or average) value.
• If we know the probability distribution of X we
  can find the expected value and variance (square
  of the standard deviation).

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Formulas

• Expected value
• ? E(X) ? Xi PrXi.
• I.e. The sum of each possible value of X times
  its probability.
• Variance
• ?2 Var(X) ? (Xi - ?)2 PrXi.
• I.e. The sum of the squared deviation from the
  mean of each possible value times its probability.

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Example

• Tossing three coins.
• X number of heads.
• PrX 0 1/8, PrX 1 3/8, PrX 2 3/8
  and PrX 3 1/8.
• E(X) 0 ? 1/8 1 ? 3/8 2 ? 3/8 3 ? 1/8
  1.5.
• Var(X) (0 - 1.5)2 ? 1/8 (1 - 1.5)2? 3/8
• (2 - 1.5)2 ? 3/8 (3 - 1.5)2 ? 1/8 0.75.
• ? ? Var(X) 0.866.

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Other distributions

• There are several other important probability
  distributions.
• This lecture has outlined two of them.
• The normal distribution is called that not
  because it is the usual case (it isnt) but
  because it is a norm (or standard).
• We will use this standard in some of the later
  topics.
• If we know the process, we might be able to
  calculate the probability distribution of all
  possible outcomes.
• E.g. In card games and other games of chance.
• If we have many observations of past outcomes,
  but we cannot calculate the probabilities from
  theory, we can test whether the outcomes conforms
  with one of the standard probability
  distributions.