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MF-852 Financial Econometrics
Lecture 4
Probability Distributions and
Intro. to Hypothesis Tests
Roy J. Epstein
Fall 2003
1
Distribution of a Random
Variable



A random variable takes on different
values according to its probability
distribution.
Certain distributions are especially
important because they describe a
wide variety of random variables.
Binomial, Normal, student’s t
2
Binomial Distribution

Random variable has two outcomes, 1
(“success”) and 0 (“failure”)


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Coin flip: heads = 1, tails = 0
P(success) = p
P(failure) = q = (1 – p)
Binomial distribution yields probability
of x successes in n outcomes.
Excel will do the calculations.
3
Tails and Body of a
Distribution
Binomial Distribution
p = 0.4, n = 8
30.0%
25.0%
Probability
20.0%
15.0%
10.0%
upper tail
5.0%
0.0%
0
1
2
3
4
5
6
7
8
Successes
4
Binomial Example (RR p. 20)


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Medical treatment has p = .25.
n = 40 patients
What is probability of at least 15
successes (cures)
I.e, P(x  15)?
5
Normal Distribution

A normally distributed random
variable:





Is symmetrically distributed around its
mean
Can take on any value from – to +
Has a finite variance
Has the famous “bell” shape
“Standard normal:” mean 0, variance
1.
6
z
3
2.85
2.7
2.55
2.4
2.25
2.1
1.95
1.8
1.65
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
0.3
0.15
0
-0.2
-0.3
-0.5
-0.6
-0.8
-0.9
-1.1
-1.2
-1.4
-1.5
-1.7
-1.8
-2
-2.1
-2.3
-2.4
-2.6
-2.7
-2.9
-3
f(z)
Standard Normal Distribution
0.450
0.400
0.350
0.300
0.250
0.200
0.150
tail area
0.100
0.050
0.000
7
3
2.85
2.7
2.55
2.4
2.25
2.1
1.95
1.8
1.65
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
0.3
0.15
0
-0.2
-0.3
-0.5
-0.6
-0.8
-0.9
-1.1
-1.2
-1.4
-1.5
-1.7
-1.8
-2
-2.1
-2.3
-2.4
-2.6
-2.7
-2.9
-3
f(z)
N(0, .5) Distribution
1.800
1.600
1.400
1.200
1.000
0.800
0.600
0.400
0.200
0.000
z
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N(0,1) Probabilities

Suppose z has a standard normal
distribution. What is:
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
P(z  1.645)?
P(z  –1.96)?
Excel will tell us!
9
N(0,1) and Standardized
Variables

Suppose x is N(12,10).

What is P(x  24.8) ?
10
Key Properties of Normal
Distribution
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
Sum of 2 normally distributed random
variables is also normally distributed.
The distribution of the average of
independent and identically distributed
NON-NORMAL random variables
approaches normality.


Known as the Central Limit Theorem
Explains why normality is so pervasive in
data
11
12
Sample Mean

Take a sample of n independent
observations from a distribution with
an unknown .


Data are n random variables x1, … xn.
We estimate the unknown population
mean with the sample mean “xbar”:
1 n
x   xi
n 1
13
Properties of Sample Mean
1 n  1 n
1 n
1
E ( x )  E  xi    E ( xi )     n
n 1  n 1
n 1
n


Sample mean is unbiased!
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Properties of Sample Mean
1
 n  1
Var ( x )  2 Var  xi   2
n
1  n


n
1 n 2 1
2
Var
(
x
)



n

1

i
n 1
n
2
n
Sample mean has variance. But the
variance is reduced with more data.
15
Null Hypothesis


“Null hypothesis” (H0) asserts a
particular value (0) for the unknown
parameter  of the distribution.
Written as H0 :  = 0


E.g., H0 :  = 5
H0 usually concerns a value of
particular interest (e.g., given by a
theory)
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Null Hypothesis

xbar is unlikely to equal 0 exactly.

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Samples have sampling error, by
definition.
Is xbar still consistent with H0 being a
true statement?

This involves a hypothesis test.
17
Hypothesis Testing
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
Hypothesis testing finds a range for 
called the confidence interval.
The confidence interval is the set of
acceptable hypotheses for , given the
available data.
H0 is accepted if the confidence
interval includes 0.
Otherwise H0 is rejected.
18
Confidence Interval


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confidence interval = xbar  allowable
sampling error
How wide should the interval be
around xbar?
Customary to use a 95% confidence
interval.


The interval will include the true  95% of
the time
Each tail probability is 2.5%.
19
Construction of Confidence
Interval

If x1, … xn are normally distributed
then xbar is normally distributed.
Then:
P(1.96 
P( x  1.96

x  0
 1.96)  95%
( / n )

n
 0  x  1.96

n
)  95%
The 95% confidence interval is
x  1.96

n
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Confidence Interval Example

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You are a restaurant manager. Burgers are
supposed to weigh 5 ounces on average. The
night shift makes burgers with a standard
deviation of 0.75 ounces.
You eat 12 burgers from the night shift and
xbar is 5.4 ounces. What is a 95%
confidence interval for the weight of the
night shift burgers?
You eat 8 more burgers that have an average
weight of 5.25 ounces. What is a 95%
confidence interval for this sample?
What is a 95% confidence interval based on
all 20 burgers?
21
Sample Variance


Usually the population variance, as
well as the mean, is unknown.
Estimate 2 with the sample variance:
n
1
2
s2 
(
x

x
)

i
n 1 1


We divide by n-1, not n.
What is the sample variance of xbar?
22
Sample Variance


Usually the population variance, as
well as the mean, is unknown.
Estimate 2 with the sample variance:
n
1
2
s2 
(
x

x
)

i
n 1 1


We divide by n-1, not n.
What is the sample variance of xbar?
23
t-distribution
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Confidence intervals use the tdistribution instead of the normal
when the variance is estimated from
the sample.
T-distribution has fatter tails than the
normal.
Confidence intervals are wider because
we have less information.
24
t
3
2.85
2.7
2.55
2.4
2.25
2.1
1.95
1.8
1.65
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
0.3
0.15
0
-0.2
-0.3
-0.5
-0.6
-0.8
-0.9
-1.1
-1.2
-1.4
-1.5
-1.7
-1.8
-2
-2.1
-2.3
-2.4
-2.6
-2.7
-2.9
-3
f(t)
t distribution (3 dof)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
25
Confidence Interval with tdistribution

You hired Leslie, a new salesperson. Leslie
made the following sales each month in the
first half:
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January — $25,000
February — $27,000
March — $29,000
April — $20,000
May — $22,000
June — $35,000
What is a 95% confidence interval for Leslie’s
monthly sales? (assume monthly sales are
normally distributed)
Suppose you knew that the standard
deviation of sales was $1,500. How would
your conclusion change?
26
Significance Levels

Assuming H0, what is the probability
that the sample value would be as
extreme as the value we actually
observed?


Alternative to confidence interval
Equal to
P( z 
x  0
) for normal variates
( / n )
x  0
P(t 
)  for t vari ates
(s / n )
27
Type 1 and Type 2 Error


Accept or reject H0 based on the
confidence interval.
Type 1 error: reject H0 when it is true.


What is probability of this?
Type 2 error: accept H0 when it is
false.

How important is this?
28