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Random SamplingRandom Sampling- - Random SamplesRandom Samples
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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
iix
nx
1
1
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Example
Shift observed 1 2 3 4 5 6 7 8 9 10Number of stoppages 2 11 6 8 6 5 10 4 8 3
3.6)384105686112(10
1
1
1
n
iix
nx
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Important
• 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
x
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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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Sample Data
0.000
0.020
0.040
0.060
0.0800.100
0.120
0.140
0.160
0.180
0.200
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Stoppages
Rela
tive
Fre
quen
cy
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]
APPROXIMATING MASS FUNCTIONS-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
But Relative frequency of a bin= Area of the corresponding rectangle
• Now the area of each rectangle represents the probability• Now we must plot the adjusted relative frequencies against the mid points of
the bins
width
frequencylativefrequencyrelativeAdjustedHeight
Re)(
width
AreafrequencyrelativeAdjustedHeight )(
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Approximating the p.d.f. (Disney)
Bins FrequencyRelative
FrequencyAdjusted Rel. Freq.(Height) Midpoint
0.76 1 0.00240 0.07994 0.7450.79 0 0.00000 0.00000 0.7750.82 0 0.00000 0.00000 0.8050.85 1 0.00240 0.07994 0.8350.88 1 0.00240 0.07994 0.8650.91 2 0.00480 0.15987 0.8950.94 23 0.05516 1.83853 0.9250.97 56 0.13429 4.39648 0.955
1 126 0.30216 10.07194 0.9851.03 120 0.28777 9.67226 1.0151.06 53 0.12710 4.23661 1.0451.09 20 0.04796 1.59872 1.0751.12 9 0.02158 0.71942 1.1051.15 4 0.00959 0.31974 1.1351.18 1 0.00240 0.07994 1.165
More 0 0.00000 0.00000Sum: 417 1 33.333
Since the width is 0.03
0.0024/0.03
(0.73+0.76)/2
Histogram function is used for normalized ratios( Rnorm)
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Approximating the p.d.f. (Disney)
Approximation of p.d.f.
0
2
4
6
8
10
12
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
Normalized ratios
Height
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Approximating the p.d.f. (Disney)
Approximation of p.d.f.
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Height
Normalized ratios
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How can random samplinghelp 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.
Random Samples, Focus
on the project
Probability, Mathematics, Tests, Homework, Computers
StockOptionPricing
Option Focus.xls (material continues) IT CClass Project
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on the project
Probability, Mathematics, Tests, Homework, Computers
StockOptionPricing
Random, Focus
The 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.
Option Focus.xls (material continues) IT CClass Project
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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.
on the project
Probability, Mathematics, Tests, Homework, Computers
StockOptionPricing
Random, Focus (material ends) Option Focus.xls IT
SAMPLE OF NORMALIZED RATIOS
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.745 0.805 0.865 0.925 0.985 1.045 1.105 1.165
normalized ratio
heig
ht
CClass Project
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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
reportWhat’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.
X
fX
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Hw11
Estimate the E(X) for the continuous random variable X, whose p.d.f., fX is shown below.
00.10.2
0.30.40.5
0 1 2 3 4 5 6 7 8 9 10
x
f X(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
Read the description of the function from the Excel menu.
Menus differ slightlyin Excel XP.
RANDBETWEEN
FUNCTION
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Simulation, Integers
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.
VLOOKUP FUNCTION
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Simulation, Integers
For our purposes, this will usually be left blank.
Value in the leftmost column of table
Location of table
Number of the column where
value is to be found
Phone Log.xls
Simulation. Simulation. Integers: page 6Integers: page 6Simulation. Simulation. Integers: page 6Integers: page 6
(material continues) IT C
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The IF function is found under the Logical submenu of the Function Wizard.
Menus differ slightlyin Excel XP.
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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
Read the description of the function from the Excel menu.
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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
r1r2r3 r20
• cnorm = $21.8700r1r2r3 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.