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Econ 140
More on Univariate Populations
Lecture 4
Lecture 4
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Today’s Plan
Econ 140
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Examining known distributions:
Normal distribution & Standard normal curve
Student’s t distribution
F distribution & c2 distribution
Note: should have a handout for today’s lecture with all
tables and a cartoon
• Brief statements about: Bivariate populations and
conditional probabilities
• Joint and marginal probabilities
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Standard Normal Curve (6)
Econ 140
• Going back to our earlier question: What is the probability
that someone earns between $300 and $400 [P(300Y
400)]?
  316.6
Z1
Z2
 2  25608
  25608  160
P(300Y 400)
300  316.6
 0.104
160
300 316.6
400  316.6
Z 400 
 0.52
160
P (0.104  Z  0)  0.0418
P (0  Z  0.52)  0.1985
P (0.104  Z  0.52)  0.0418  0.1985  .2403
Z300 
Lecture 4
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Standard Normal Curve (7)
Econ 140
• We know from using our PDF that the chance of someone
earning between $300 and $400 is around 23%, so 0.24 is
a good approximation
• Now we can ask: What is the probability that someone
earns between $253 and $316?
Z1
Z2
P(253Y 316)
253  316.6
 0.3975
160
316  316.6
Z2 
 0.0038
160
P (0.3975  Z  0)  0.1554
Z1 
P (0.0038  Z  0)  0.0020
P (0.3975  Z  .0038)  .1554  .002
Lecture 4
253
316.6
316
 .1574  15.3%
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Standard Normal Curve (8)
Econ 140
• There are instructions for how you can do this using Excel:
L4_1.xls. Note how to use STANDARDIZE and
NORMDIST and what they represent
• Our spreadsheet example has 3 examples of different
earnings intervals, using the same distribution that we used
today
• Testing the Normality assumption. We know the
approximate shape of the Earnings (L3_79.xls)
distribution. Slightly skewed. Is normality a good
assumption? Use in Excel (L4_2.xls) of NORMSINV
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Student’s T-Distribution
Econ 140
• Starting next week, we’ll be looking more closely at
sample statistics
• In sample statistics, we have a sample that is small relative
to the population size
• We do not know the true population mean and variance
– So, we take samples and from those samples we will
estimate a mean Y and variance SY2
Lecture 4
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T-Distribution Properties
Econ 140
• Fatter tails than the Z distribution
• Variance is n/(n-2) where n is the number of observations
• When n approaches a large number (usually over 30), the t
approximates the normal curve
• The t-distribution is also centered on a mean of zero
• The t lets us approximate probabilities for small samples
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F and c2 Distributions
Econ 140
• Chi-squared distribution:square of a standard normal (Z)
distribution is distributed c2 with one degree of freedom
(df).
• Chi-squared is skewed. As df increases, the c2
approximates a normal.
• F-distribution: deals with sample data. F stands for Fisher,
R.A. who derived the distribution. F tests if variances are
equal.
• F is skewed and positive. As sample sizes grow infinitely
large the F approximates a normal. F has two parameters:
degrees of freedom in the numerator and denominator.
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A recap on the story so far
Econ 140
• Probability is concerned with random events.
• Nearly all data is the outcome of a ‘random draw’ - a
sample drawn at random.
• The probability of earning particular amounts
– Relationship between a sample and population
– Using standard normal tables
• Introduction to the t-distribution
• Introduction to the F and c2 distributions
Lecture 4
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A quick note on bivariate probability
Econ 140
• Bivariate populations and conditional probabilities
• Joint and marginal probabilities
Lecture 4
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A Simple E.C.P Example
Econ 140
• Introduce Bivariate probability with an example of
empirical classical probability (ecp).
• Consider a fictitious computer company. We might ask the
following questions:
– What is the probability that consumers will actually buy
a new computer?
– What is the probability that consumers are planning to
buy a new computer?
– What is the probability that consumers are planning to
buy and actually will buy a new computer?
– Given that a consumer is planning to buy, what is the
probability of a purchase?
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A Simple E.C.P Example(2)
Econ 140
• Think of probability as relating to the outcome of a random
event (recap)
• All probabilities fall between 0 and 1:
null 0  P( A)  1 certain
• Probability of any event A is:
m
P( A)  with A  a1, a2 , a3...an 
n
Where m is the number of events A and n is the number
of possible events
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A Simple E.C.P Example(3)
Econ 140
• The cumulative frequency is:  P(ai )  1
• The sample space (of a 1000 obs) looks like this:
Plan
to Purchase
Yes (a1)
No (a2)
Total
Actually Purchase
Yes (b1) No (b2) Total
200
50
250
100
650
750
300
700
1000
• Before we move on we’ll look at some simple definitions
Lecture 4
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A Simple E.C.P Example(4)
Econ 140
• If we have an event A there will be a compliment to A
which we’ll call A’ or B
• Computing marginal probabilities
– Event A consists of two outcomes, a1 and a2:
A  a1,a2 
– The compliment B consists also of two outcomes, b1
and b2:
B  b1,b2 
– two events are mutually exclusive if both events
cannot occur
– A set of events is collectively exhaustive if one of the
events must occur
Lecture 4
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A Simple E.C.P Example(5)
Econ 140
• Computing marginal probabilities
Pr( A)  P A  B1   P A  B2   ...  P A  Bk 
Where k is some arbitrary large number
• If A = planned to purchase and B=actually purchased:
P(planned to buy) = P(planned & did) + P(planned & did not)=
Plan
to Purchase
Yes (a1)
No (a2)
Total
Actually Purchase
Yes (b1) No (b2) Total
200
50
250
100
650
750
300
700
1000
200
50
250


 0.25
1000 1000 1000
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A Simple E.C.P Example(6)
Econ 140
• If the two events, A and B, are mutually exclusive, then
P( AorB)  P( A)  P( B)
– General rule written as:
P( AorB)  P( A)  P( B)  P( A  B)
– Example: Probability that you draw a heart or spade
from a deck of cards
• They’re mutually exclusive events
P(Heart or Spade) = P(Heart) + P(Spade) – P(Heart + Spade)=
13 13
26 1
 0 
  0.50
52 52
52 2
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A Simple E.C.P Example(6)
Econ 140
• Probability that someone planned to buy or actually did
buy: use the general addition rule:
P( AorB)  P( A)  P( B)  P( A  B)
• If A is planning to purchase, and B is actually purchasing,
we can plug in the marginal probabilities to find
250 300 200 350



 0.35
1000 1000 1000 1000
Plan
to Purchase
Yes (a1)
No (a2)
Total
Actually Purchase
Yes (b1) No (b2) Total
200
50
250
100
650
750
300
700
1000
Joint Probability: P(A and B): Planned and Actually Purchased
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Conditional Probabilities
Econ 140
• Lets leave the example for a while and consider
conditional probabilities.
• Conditional probabilities are represented as P(Y|X)
• This looks similar to the conditional mean function:
Y X
n
• We’ll use this to lead into regression line inference.
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Conditional Probabilities (2)
Econ 140
• Probabilities will be defined as
p jk  P( X  X j , Y  Yk )
j  1,...J
k  1,...K
• If we sum over j and k, we will get 1, or:
 j k p jk  1
• We define the conditional probability as f (X|Y)
– This is read “a function of X given Y”
– We can define this as:
Joint probability of X &Y
 f X | Y 
Marginal probability of Y
Lecture 4
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Conditional Probabilities (3)
Econ 140
• Similarly we can define f (Y|X):
Joint probability of Y &X
 f Y | X 
Marginal probability of X
• Looking at our example spreadsheet, we have a sample of
weekly earnings and years of education: L5_1.XLS.
• There are two statements on the spreadsheet that will
clarify the difference between a joint and conditional
probabilities
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Conditional Probabilities (4)
Econ 140
• The joint probability is a relative frequency and it asks:
– How many people earn between $600 and $799 and
have 10 years of education?
• The conditional probability asks:
– How many people earn between $600 and $799 given
they have 10 years of education?
• On the spreadsheet I’ve outlined the cells that contain the
highest probability in each completed years of education
– There’s a pattern you should notice
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Conditional Probabilities (5)
Econ 140
• We can use the same data to graph the conditional mean
function
– the graph shows the same pattern we saw in the
outlined cells
– The conditional probability table gives us a small
distribution around each year of education
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Conditional Probabilities (6)
Econ 140
• To summarize, conditional probabilities can be written as
P( X &Y ) Joint probability of X & Y

 f (X |Y)
P( X )
Marginal probability of X
– This is read as “The probability of X given Y”
– For example: The probability that someone earns
between $200 and $300, given that he/she has
completed 10 years of education
• Joint probabilities are written as P(X&Y)
– This is read as “the probability of X and Y”
– For example: The probability that someone earns
between $200 and $300 and has 10 years of education
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A Marketing Example
Econ 140
• Now we’ll look at joint probabilities again using the
marketing example from earlier in the lecture.
• We will look at:
– Marginal probabilities P(A) or P(B)
– Joint probabilities P(A&B)
– Conditional probabilities P( A& B)
P( B)
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Marketing Example(2)
Econ 140
• Here’s the matrix
Plan
to Purchase
Yes
No
Total
Actually Purchase
Yes
No
Total
200
50
250
100
650
750
300
700
1000
• Let’s look at the probability you purchased a computer
given that you planned to purchase:
• P(actually purchased | planned to purchase)  200  .8  80%
250
• The joint probability that you purchased and planned to
purchase: 200/1000 = .2 = 20%
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What we’ve done
Econ 140
• Introduction to standardized normal (Z) distribution
• Introduction to the t-distribution
• Introduction to the F and c2 distributions
• Bi-variate probabilities: calculate marginal, joint, and
conditional probabilities
– Computer company example
– Earnings and years of education
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