Probability%20and%20Probability%20Distributions

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Probability and Probability Distributions

• ASW, Chapter 4-5
• Skip sections 4.5, 5.5, 5.6

September 24, 2008
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Conditional probabilities (ASW, 162-167)

• A conditional probability refers to the
  probability of an event A occurring, given that
  another event B has occurred.
• Notation P(A ? B)
• Read this as the conditional probability of A
  given B or the probability of A given B.
• Conditional probabilities are especially useful
  in economic analysis because probabilities of an
  event differ, depending on other events occurring.

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Formulae for conditional probabilities

• The conditional probability of A given B is
• The conditional probability of B given A is

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Number of students by major and Excel skill level
Major of student Excel skill level Excel skill level Excel skill level Excel skill level Total
Major of student None (N) Low (L) Medium (M) High (H) Total
Math (MA) 0 2 4 0 6
Business (B) 1 3 6 3 13
Economics (E) 2 12 8 2 24
Other (O) 0 1 1 1 3
Total 3 18 19 6 46
This table contains the same data as examined
earlier, but reorganized as a table rather than
in a tree diagram.
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Examples of conditional probabilities from
student survey

• Probability that each major has low skill level?
• P(L ? MA) P(L ? MA) / P(MA) (2/46) / (6/46)
  2/6 0.333
• P(L ? B) 3 / 13 0.231
• P(L ? E) 0.500
• P(L ? O) 0.333
• If a student has a high skill level is Excel,
  what is the probability his or her major is
  Business? Other?
• P(B ? H) P(B ? H) / P(H) (3/46) / (6/46)
  3/6 0.500
• P(O ? H) 0.167

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Using conditional probabilities

• While four-day workweeks make some sense for the
  manufacturing sector, its much more challenging
  for service-based companies that have to be
  available for clients questions. Even on
  Fridays.
• Source The Globe and Mail, September 20, 2008,
  B18.
• Parents who resided in the largest census
  metropolitan areas were more likely to have an
  adult child at home. For example, 41 of parent
  in Vancouver but only 17 of parent living in
  rural areas or small towns shared their house
  with at least one adult child.
• Source Parents with adult children living at
  home. Statistics Canada, Canadian Social
  Trends, Spring 2006.

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Sample of Saskatchewan residents

• Random sample of 2,500 Saskatchewan residents
  from the Census of Canada, 2001 Public Use
  Microdata File, Individuals File. Obtained from
  the Internet Data Library System, through the
  University of Regina Data Library Services.
• Subgroup selected was those with ages 30-64
  years, wages and salaries greater than zero, and
  full-time jobs in the year 2000.
• This resulted in a sample of 700 individuals.

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Number of Saskatchewan residents with various
levels of wages and salaries and schooling, 2000
Wages and salaries Years of schooling Years of schooling Years of schooling Years of schooling Total
Wages and salaries lt12 12-13 14-17 18 Total
lt20,000 38 84 47 8 177
20-45,000 69 135 101 20 325
45,000 21 72 82 23 198
Total 128 291 230 51 700
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Some conditional probabilities

• What is the conditional probability of 45,000 in
  wages and salaries given less than twelve years
  of schooling? Given 14-17 years of schooling?
  Given 18 years?
• P(45 ? lt12) P(45 ? lt12) / P(lt12) 21/ 128
  0.164
• P(45 ? 14-17) 82/ 230 0.357
• P(45 ? 18) 0.451
• That is, chances of a high income increase with
  each higher level of schooling.
• What is the probability that someone with a
  middle level of income has 12-13 years of
  schooling?
• P(12-13 ? 20-45) P(12-13?20-45) / P(lt20-45)
  135/ 325 0.415

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Conditional probabilities of various levels of
wages and salaries, given years of schooling,
n700 Saskatchewan residents, 2000
Wages and salaries Years of schooling Years of schooling Years of schooling Years of schooling Total
Wages and salaries lt12 12-13 14-17 18 Total
lt20,000 0.297 0.289 0.204 0.157 0.253
20-45,000 0.539 0.464 0.439 0.392 0.464
45,000 0.164 0.247 0.357 0.451 0.283
1.000 1.000 1.000 1.000 1.000
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Organizing cross-classification tables

• ASW use joint probability tables with joint and
  marginal probabilities. Study the example on
  pages 163-164.
• Joint probabilities are the probabilities of the
  intersection of each pair of events in a
  cross-classification table.
• Marginal probabilities are the probabilities of
  each of the events in the rows and columns of the
  table.
• Conditional probabilities can be computed from
  the numbers of cases, as reported in the
  cross-classification table, as in the examples
  shown above. I find this method more useful for
  the following analysis of independence and
  dependence.

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Independent and dependent events (ASW, 166)

• Two events A and B are independent if
• P(A ? B) P(A) or P(B ? A) P(B).
• That is, the probability of one event is not
  altered by whether or not the other event occurs.
  
• If P(A ? B) P(A), then P(B ? A) P(B), and
  vice-versa.
• Two events A and B are dependent if
• P(A ? B) ? P(A) or P(B ? A) ? P(B).
• In this case, the occurrence of one event affects
  the probability of the other event.

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Example of dependence

• Does the event of having low wages and salaries
  depend on having few years of schooling?
• If A is the event of having a low salary
  (lt20,000) and B is the event of having less than
  twelve years of schooling
• P(A ? B) 38/128 0.297
• P(A) 177/700 0.253
• And P(A ? B) gt P(A) so the chance of having low
  wages and salaries is greater for those with the
  least amount of schooling, as compared with the
  whole sample.
• Also note in this case that P(B ? A) 38/177
  0.215 gt 0.183 P(B). This is an alternative
  way of checking for whether the events are
  dependent or independent.

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Example of independence

• Are the events of having 12-13 years of schooling
  (A) and the event of having wages and salaries of
  20-45,000 (B) dependent or independent?
• P(A ? B) 135/325 0.415
• P(A) 291/700 0.416
• So these two events are essentially independent
  of each other. Also note that
• P(B ? A) 135/291 0.464
• P(B) 325/700 0.464
• In this case, those with a middle level of
  schooling (12-13 years) and the middle category
  of income are similar to a cross-section of the
  whole sample.

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Using dependence and independence

• Some authors have argued that parents in higher
  socio-economic positions may have a greater
  tendency to expect their children to be
  independent earlier than those with less
  education and income.However, the analysisdoes
  not show support for these interpretations.
  Parents with a higher level of education were
  neither more not less likely than less
  well-educated parents to live with their adult
  children. Nor were parents with high personal
  income any less likely than those with lower
  personal income to provide accommodation for
  their children.
• Source Parents with adult children living at
  home. Statistics Canada, Canadian Social
  Trends, Spring 2006.

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Independence and dependence in economic analysis

• Is the price of wheat received by Saskatchewan
  farmers dependent on the weather in Russia?
• Is the chance of NAFTA being renegotiated
  dependent on the result of the U.S. presidential
  election?
• Is the consumption of table salt dependent on
  interest rates? Is it dependent on health fads?

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Multiplication rule (ASW, 165)

• The multiplication rule can be used to compute
  the probability of the intersection of two
  events.
• P(A n B) P(A) P(B ? A)
• P(A n B) P(B) P(A ? B)
• But note that if events A and B are independent
  of each other, then P(B ? A) P(B) and P(A ? B)
  P(A), so that
• P(A n B) P(A) P(B)

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Example of multiplication rule

• What is the probability of wages and salaries of
  20-45,000 (A) and having 12-13 years of
  schooling (B)?
• Since we already know that A and B are
  independent,
• P(A) x P(B) (325/700) x (291/700) 0.193.
• Note that P(A n B) 135 / 700 0.193 from the
  table.
• What is the probability of wages and salaries of
  45,000 (C) and having 14-17 years of schooling
  (D)?
• In this case, we have not checked for
  independence, so use the full formula
• P(C n D) P(C) P(D ? C) (198/700)X(82/198)
  82/700 0.117

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Using independence

• Independent trials of an experiment
• Successive flips of a coin.
• Many rolls of a die or a pair of dice.
• Sale of a product to customers arriving at a
  retail store.
• If a population is small and a case that is
  selected is not replaced before the next case is
  drawn, then successive drawings are dependent on
  each other. But if the population is large,
  successive draws do not alter the composition of
  the population. Thus, random selection of
  respondents from a large population produces
  independence of successive selections.
• When trials of an experiment are independent of
  each other, then the binomial probability
  distribution can be used to determine the
  probability of several occurrences of an event in
  many trials ASW, section 5.4.

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Random variables (ASW, 185)

• A random variable is a numerical description of
  the outcome of an experiment.
• Or, a random variable attaches a numerical value
  to each possible experimental outcome.
• A random variable is often assigned an algebraic
  symbol such as x.
• A random variable can be either discrete
  (countable number of possible values) or
  continuous (not countable or any numerical value
  with an interval).
• Chapter 5 deals with discrete random variables.
• Chapter 6 deals with continuous random variables.

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Discrete random variables

• Any random variable that has a finite number of
  possible values or a countably infinite number of
  possible values.
• Examples
• The number of females in a sample of 3 persons
  selected from a large population that is half
  female and half male (x 0, 1, 2, 3).
• The sum of the faces shown when a pair of dice is
  rolled (x 2, 3, 4, , 12).
• The number of customers at a restaurant at lunch
  (x 0, 1, 2, 3, 4, 5, , 45). To the maximum
  of the number of seats.
• The number of unemployed workers in Saskatchewan
  reported by Statistics Canada each month (x 0,
  1, 2, , 29,800).
• The number of homeowners who have defaulted on
  mortgages in the United States during the last
  year.

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Continuous random variables

• Any random variable whose possible values cannot
  be counted is termed continuous. Alternatively,
  if the possible outcomes can take on any
  numerical value in an interval or set of
  intervals, the random variable is continuous.
• Examples
• Number of kilometres goods are transported from a
  manufacturing plant to a warehouse.
• Time taken to ship the goods.
• Exchange rates for currencies.
• Household income.
• We will study the continuous uniform distribution
  and the normal distribution (bell curve) next
  week.

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Probability distributions

• A probability distribution is a random variable,
  along with the associated probabilities of
  occurrence of the values of the variable.
• Discrete probabilities of each value of the
  random variable.
• Continuous probability that the random variable
  is within a particular interval.

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Discrete probability distribution. (ASW, 189-192)

• For a discrete random variable x, the probability
  distribution is the set of values of x, along
  with f(x), the function that gives the
  probability for each value of x.
• For each value of x, f(x) is no less than 0 and
  no greater than 1. The sum of the probabilities
  for all values of x equals 1. Symbolically,
• 0 ? f(x) ? 1
• ? f(x) 1

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Probability distribution for sex of person
selected
Equally likely outcomes for experiment of
randomly selecting 3 persons from a large
population of half males and half
females FFF FFM FMF FMM MFF MFM MMF MMM
Let the random variable x be the number of
females selected and f(x) the probabilities for
each value of x.

x f(x)
0 1/8 0.125
1 3/8 0.375
2 3/8 0.375
3 1/8 0.125
Total 8/8 1.000
In this example, the values of f(x) are obtained
using the classical interpretation of probability.
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Responses to Would you like to lower tuition,
even if it meant larger class sizes?
Response Numerical value Number of respondents Relative frequency
Strong no 1 2 0.044
Weak no 2 5 0.111
Indifferent 3 10 0.222
Weak yes 4 8 0.178
Strong yes 5 20 0.444
Total Total 45 0.999
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Probability distribution and expected value for
lower tuition question
If a student is randomly selected, let x be the
response to the lower tuition question. In this
case, the values of the probability function f(x)
are the relative frequencies of occurrence of the
responses to the question.
x f(x) xf(x)
1 0.044 0.044
2 0.111 0.222
3 0.222 0.666
4 0.178 0.712
5 0.445 2.225
Total 1.000 E(x) 3.869
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Graphing discrete probability distributions

• Use a line chart as in Figure 5.1 of ASW. Or it
  could be a bar chart with spaces left between the
  bars, to visually indicate that it is a discrete
  distribution.
• Convention is to place the values of the random
  variable x on the horizontal axis and values of
  the probability function f(x) on the vertical
  axis.
• Examples that follow illustrate these methods.

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Source Fall 2005 Survey, prepared by Harvey King
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Expected values (ASW, 195)

• The expected value E(x) of a random variable x is
  the mean of the probability distribution.
  Symbolically,
• E(x) µ ? x f(x)
• where µ (pronounced something like mu) is a
  Greek symbol used to indicate mean.
• The concept of expected value is more general
  than just referring to the mean, in that the
  expected value can be obtained for other
  expressions see later notes on the variance.
  However, in this course, it will be used to
  denote the expected value of x, or the mean.

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Expected value for x, number of females selected

x f(x) x f(x)
0 1/8 0.125 0.000
1 3/8 0.375 0.375
2 3/8 0.375 0.750
3 1/8 0.125 0.375
Total 8/8 1.000 1.500
E(x) µ ? x f(x) 1.500
If a random sample of 3 persons is obtained from
a large population composed of half females and
half males, the expected number of females
selected is 1.5. If there are many samples of 3
persons each time, the mean number of females
across the samples is 1.5.
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Expected value for responses to lower tuition
question
The expected value of the responses is 3.869, or
3.9. Recall that a response of 3 was
indifferent and a response of 4 was weak yes
so, in this sample, the expected value or mean is
just below weak yes.
x f(x) xf(x)
1 0.044 0.044
2 0.111 0.222
3 0.222 0.666
4 0.178 0.712
5 0.445 2.225
Total 1.000 E(x) 3.869
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Variance (ASW, 195)

• The variance of a probability distribution is the
  expected value of the squares of the differences
  of the random variable x from the mean µ.
  Symbolically,
• Var(x) s2 ?(x µ)2 f(x)
• The Greek symbol s is sigma.
• The variance can be difficult to calculate and
  interpret. It is in units that are the square of
  the random variable x. Partly because of this,
  in statistical work it is more common to use the
  square root of the variance or s. The standard
  deviation has the same units as x.

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Variance of x, number of females selected

x f(x) x f(x) x - µ (x µ)2 (x µ)2f(x)
0 1/8 0.125 0.000 -1.5 2.25 0.28125
1 3/8 0.375 0.375 -0.5 0.25 0.09375
2 3/8 0.375 0.750 0.5 0.25 0.09375
3 1/8 0.125 0.375 1.5 2.25 0.28125
Total 8/8 1.000 1.500 0.75000
If a random sample of 3 persons is obtained from
a large population composed of half females and
half males, the expected number of females
selected is µ 1.5. The variance of the number
of females selected is Var(x) s2 ?(x µ)2
f(x) 0.75. The standard deviation is the
square root of 0.75, so that s 0.866.
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Later this class or next day

• Binomial probability distribution
• Continuous probability distributions
• Bring along copies of the Normal Distribution for
  Monday and Wednesday, Sept. 29 and October 1.
  This is Table 1 of Appendix B of ASW.