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 - 2 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. 3 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 4 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 5 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 6 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). 7 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] 8 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 9 Important But we know Area of rectangle=width x height Height ( Adjusted But Relative frequency of a bin= Area of the corresponding rectangle Height ( Adjusted relative Area frequency )

width relative 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 10 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

Frequency 0.76 1 0.79 0 0.82 0 0.85 1 0.88 1 0.91 2 0.94 23 0.97 56 1 126 1.03 120 1.06 53 1.09

20 1.12 9 1.15 4 1.18 1 More 0 Sum: 417 Relative Frequency 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) Midpoint 0.07994 0.745 0.00000 0.775 0.00000 0.805 0.07994 0.835 0.07994 0.865 0.15987 0.895 1.83853

0.925 4.39648 0.955 10.07194 0.985 9.67226 1.015 4.23661 1.045 1.59872 1.075 0.71942 1.105 0.31974 1.135 0.07994 1.165 0.00000 33.333 (0.73+0.76)/2 11

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 12 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 13 Normalized ratios Stock Probability, Mathematics,

Tests, Homework, Computers Option 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. Random Samples, Focus Option Focus.xls Class Project

(material continues) T C I 14 Stock Probability, Mathematics, Tests, Homework, Computers Option 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, Option Focus.xls Class Project Focus (material continues) T C I 15 SAMPLE OF NORMALIZED RATIOS 12.0 10.0

height 8.0 on the project 6.0 Stock Probability, Mathematics, Tests, Homework, Computers Option 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) T C I 16 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 Whats 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 17 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 18 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 19 Bootstrapping collect a smaller sample of data points, then use a computer to simulate a much

larger set. 20 Computer Simulations Why do simulations? Cost benefits Time constraints Availability of data Which Excel functions will we need? RANDBETWEEN() discrete case - VLOOKUP() IF() 21 Simulation, Integers RANDBETWEEN FUNCTION Menus differ slightly

in Excel XP. Read the description of the function from the Excel menu. 22 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. 23

Simulation, Simulation. Integers: page page66 Integers Integers: Simulation. 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) T

C I 24 Menus differ slightly in Excel XP. The IF function is found under the Logical submenu of the Function Wizard. 25 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

26 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. 27 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 28

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. 29 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, 30 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. 31 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. 32