Transcript Random Sampling - Random Samples

Random Sampling
- Random Samples
1
Why do we need Random
Samples?
• Many business applications
-We will have a random variable X such that
the probability distribution & expected
value is unknown
-The only way to make use of probability is
to estimate E(X) and if possible Fx or fx
-This
can be done with random sampling
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Random Samples
X come from some random process.
x results from a trial of the process
(observation of X )
a set {x1, x2, , xn} of n independent
observations of the same random variable
X is called a random sample of size n.
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Sample Mean
• What does a random sample tell us about
a random variable?
• Consider random sample
set {x1, x2, , xn}
SAMPLE MEAN
n
1
x    xi
n i 1
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Example
Shift observed
Number of stoppages
1
2
2
11
3
6
4
8
5
6
6
5
7
10
8
4
9
8
10
3
1 n
x    xi
n i 1
1
  (2  11  6  8  6  5  10  4  8  3)  6.3
10
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Important
•
x
can be used as an estimate of the
parameter E(X).
• In general, the larger the sample size n,
the better will be the estimate
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APPROXIMATING MASS AND
DENSITY FUNCTIONS
If we have a large enough sample, we can group
the data and form a histogram that approximates
• the probability mass function
(for a finite random variable)
or
• the probability density function
(for a continuous random variable).
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APPROXIMATING MASS FUNCTIONS
-Discrete random variables
Relative Frequency
Sample Data
0.200
0.180
0.160
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14
Stoppages
We have used bins of width 1 and have plotted relative frequencies.
The relative frequency of each value of X in the sample gives an
estimate for the probability that X will assume that value.
Hence, the relative frequency of a value x in the sample
approximates P(X = x) = fX(x) [p.m.f - Discrete random variables]
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APPROXIMATING DENSITY
FUNCTION-Continuous random variables
• Recall that the p.d.f can be used to find
probabilities P(a  X  b) is equal to the
area under the curve of the p.d.f over the
interval [a,b]
• If we want to use a histogram approximate
the p.d.f then
Relative frequency of a bin= Area of the corresponding
rectangle
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Important
• But we know
Area of rectangle=width x height
Height ( Adjusted relative
But
Relative frequency of a bin= Area of the corresponding
rectangle
Height ( Adjusted relative
•
•
Area
frequency) 
width
Re lative frequency
frequency) 
width
Now the area of each rectangle represents the probability
Now we must plot the adjusted relative frequencies against the mid points of
the bins
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Histogram
function is
used for
normalized
ratios( Rnorm)
Approximating the p.d.f.
Since the width is 0.03
(Disney)
0.0024/0.03
Bins
0.76
0.79
0.82
0.85
0.88
0.91
0.94
0.97
1
1.03
1.06
1.09
1.12
1.15
1.18
More
Sum:
Relative
Frequency Frequency
1
0
0
1
1
2
23
56
126
120
53
20
9
4
1
0
417
0.00240
0.00000
0.00000
0.00240
0.00240
0.00480
0.05516
0.13429
0.30216
0.28777
0.12710
0.04796
0.02158
0.00959
0.00240
0.00000
1
Adjusted Rel.
Freq.(Height)
0.07994
0.00000
0.00000
0.07994
0.07994
0.15987
1.83853
4.39648
10.07194
9.67226
4.23661
1.59872
0.71942
0.31974
0.07994
0.00000
33.333
Midpoint
0.745
0.775
0.805
0.835
0.865
0.895
0.925
0.955
0.985
1.015
1.045
1.075
1.105
1.135
1.165
(0.73+0.76)/2
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Approximating the p.d.f.
(Disney)
Approximation of p.d.f.
Height
12
10
8
6
4
2
0
0.76 0.79 0.82 0.85 0.88 0.91 0.94 0.97
1
1.03 1.06 1.09 1.12 1.15 1.18
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Normalized ratios
Approximating the p.d.f.
(Disney)
Approximation of p.d.f.
Height
12
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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Normalized ratios
Stock
Probability, Mathematics, Option
Tests, Homework, Computers
Pricing
on the project
How can random sampling
help us price a stock option?
Recall that the continuous random
variable Rnorm gives the normalized ratio of
weekly closing prices on Walt Disney stock.
The sheet Sample of the file Option Focus.xls
computes 417 values of this normalized ratio
from our 417 weekly closing ratios. We will
assume that these are independent observations
of Rnorm, that constitute a random sample of
size 417 for Rnorm.
Option Focus.xls
Random Samples, Focus
Class Project
(material continues)
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Stock
Probability, Mathematics, Option
Tests, Homework, Computers
Pricing
on theTheproject
sample mean, 1.0007695, of these observations is computed
in the sheet Sample of Option Focus.xls. This is the same as our estimate
for the ratio, Rrf, which corresponds to the weekly risk-free interest rate.
Since we constructed the normalized ratios to make this true, we have a
check on the correctness of our work.
The HISTOGRAM function is used to group the sample data for
Rnorm, and a plot is created with the total area for all of the bars being
equal to 1. This produces the bar graph, shown in the sheet Sample,
which approximates the p.d.f., fnorm, of Rnorm. Connecting the midpoints at
the tops of the bars produces the line graph approximation for fnorm that is
shown in Sample.
Our plots, which are also shown on the next page, give a visual
indication for the volatility of Disney stock over the past 8 years.
Random,
Focus
Option
Focus.xls
Class Project
(material continues)
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SAMPLE OF NORMALIZED RATIOS
12.0
10.0
height
8.0
on the project
6.0
Stock
Probability, Mathematics, Option
Tests, Homework, Computers
Pricing
4.0
2.0
0.0
0.745 0.805 0.865 0.925 0.985 1.045 1.105 1.165
normalized ratio
WHAT SHOULD YOU DO?
Each team should now plot an approximation of the probability
density function for the normalized ratios of weekly closing prices for its
particular stock data and should find the sample mean of the normalized
ratios.
Random,
Focus
Option
Focus.xls
Class Project
(material ends)
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Important
• Check class website
• Where you should be with the Project
–
–
–
–
Normalize the ratios of closing prices
Create a histogram of normalized ratios
Read the Requirements for the Project 2 written report
Write a draft of the Discussion of Options section of written
report
What’s ahead?
– Incorporate your simulation results for RANDBETWEEN in the
Discussion of Simulation section
– Decide which assumption that you will discuss in more detail in
the written report
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Expected Value
To consider the expected value (or mean) of a continuous
random variable, we can use the probability density
function(p.d.f) to give us a geometric interpretation. The
expected value perfectly “balances” the area to its right with
the area to its left.
fX
X
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Hw11
f X (x)
Estimate the E(X) for the continuous random
variable X, whose p.d.f., fX is shown below.
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
x
E(X)=4 (balances the area to its right with the area to its left
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Bootstrapping
• collect a smaller sample of data points,
then use a computer to simulate a much
larger set.
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Computer Simulations
• Why do simulations?
– Cost benefits
– Time constraints
– Availability of data
• Which Excel functions will we need?
– RANDBETWEEN() – discrete case
- VLOOKUP()
– IF()
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Simulation, Integers
RANDBETWEEN
FUNCTION
Menus differ slightly
in Excel XP.
Read the description
of the function from the
Excel menu.
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Simulation, Integers
VLOOKUP FUNCTION
This process uses
the function VLOOKUP,
which is found in the
Lookup & Reference
submenu of the Function
Wizard.
Read the description
of the function from the
Excel menu.
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Simulation, Integers
Simulation.
Integers: page 6
Value in the leftmost
column of table
Location of table
Number of the
column where
value is to be found
For our purposes, this will
usually be left blank.
Phone Log.xls
(material continues)
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Menus differ slightly
in Excel XP.
The IF function is found under
the Logical submenu of the Function
Wizard.
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Read the description of the
function from the Excel menu.
If B12 is less than or equal to 0.5, the function returns an H. If B12
is not less than or equal to 0.5, a T is returned. Note that the desired text
must be specified in quotation marks. For the stock option project we do not
need quotation marks-will show later in excel
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More on Project Focus
• r1 -a normalized ratio of the adjusted closing price at the
end of the first week (will be selected randomly)
•
initial price - $21.8700
• r1 is a value of Rnorm that might have occurred for the
first week.
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• Likewise, let r2, r3, , r20 be normalized ratios of
adjusted closing prices for weeks 2, 3, , 20 of the option.
•
It is stated in the project description that observations of R
are all independent. Thus, the normalized ratios r1, r2, r3 ,
 , and r20 are all independent observations of Rnorm
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• random observation for the ratio
r1r2r3  r20
•
cnorm = $21.8700r1r2r3  r20
•
cnorm is a observation of normalized closing price, Cnorm.
• We conclude that any set of 20 observations of R
determines a set of 20 observations of Rnorm and, therefore,
an observation of Cnorm.
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•
•
•
FV - the continuous random variable giving the
per share value of our Walt Disney call at the end of
twenty weeks, based on the normalized closing price.
s0 - strike price of $23
Cnorm takes on a value cnorm,
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Recall
• if closing stock price strike price
• Final value of Call option (Intrinsic Value of a Call)
=Maximum of 0 and C – S
• Case 1. FV = cnorm  s0 if cnorm  s0,
• Case2. FV=0
if cnorm < s0.
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• 20 observations of Rnorm determine an
observation of Cnorm, which, together with the
strike price, determines an observation of FV.
• PV - the continuous random variable giving the
present value of the Disney option on January
11, 2002.
• The present value for a value of FV is an
observation of PV. Our price for the present
value of the option is the expected value of
PV.
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