SPREADSHEET SIMULATION OF THE LAW OF LARGE NUMBERS

INTRODUCTION

If larger and larger samples are successively drawn from a population and a running average calculated after each sample has been drawn, the succession of averages will converge to the mean, ?? of the population. This remarkable fact, known as the law of large numbers, clinchs true if we draw samples from a population of discrete or continuous values, or draw them from a population with a skewed distribution.

The intention of this article is to near an Excel workbook to use in an undergraduate introductory statistics class to simulate an application of the law of large numbers. The put togethered workbook spreadsheets use standard outvie functions and do not require any higher horizontal programming. Once built, the workbook can be reused for another application of the law of large numbers.

To use the spreadsheet simulation, a user pierces a discrete probability distribution that simulates the possible issues of any random process. The spreadsheet calculates and displays the population mean, ?? that is considered to be an average number from sum of two units perspectives. First it is an wait fored value - the average of the possible values that can offer weighted by their probability of occurring. It may also be viewed as the drawn out run average of many independent observations. The simulation demonstrates this next to the first view of the mean by means of drawing many random samples from the discrete probability distribution and tracking a running average. The law of large numbers guarantees that the running average will draw gradually together to ?? , as the number of samples drawn increases.

A simulation of the random proces begins by means of using the mouse to stir a scroll bar on the chart sheet. Each time the roll of paper [i]or[/i] parchment bar is moved to the right, a sample item is drawn from the discrete distribution. The value of the sample item, a running average of all sample items, and the population wait fored value ?? are graphically displayed upon a chart. New simulation races are made by sliding the schedule bar left to clear the last move swiftly and then moving it to the right again. scholars see that different simulation races create different paths from left to right in the graph, on the contrary the law of large numbers says that whatever path we secure the long run average will always stool to the level of the population mean.

SIMULATION EXAMPLE

Suppose we are interested in simulating the arrival of customers to a fast fare restaurant during a lunch period. Assume that a Poisson random variable with mean ?? = 4 describes the number of customer arrivals through minute. Figure 1 shows the Poisson probability distribution come intoed by a user into rounded pillars A and B as value and probability pairs, and Figure 3 graphically displays the arises of a simulation run for this example. The graph displays the number of customers arriving each minute, labeled Individual Items, the running average number of customers who have visited the restaurant, and a horizontal line showing the population mean, ?? = 4 For this example, if we were to note the arrivals to the restaurant, we would diocese the fluctuating number of arrivals each minute similar to the plat of the Individual Items in the graph in Figure 3 We may not notice that, above the long run, the average number of arrivals for minute is quite stable and tend to the same points to the population mean, ?? as shown in the piece of ground of the Running Average.

BUILDING THE SIMULATION SHEETS

The simulation requires sum of two units spreadsheets and a chart sheet in a workbook named LargeNumSim.xls. The first spreadsheet, named Simulation and shown in Figure 1 is used to specify a discrete probability distribution. The probability values in rounded pillar B are entered as numerical probability values, or they may be calculated using an surpass formula, as shown in Table 2 for the above example. Tables 1 and 2 exhibit the named formulas and worksheet functions used in the Simulation spreadsheet.

The next to the first spreadsheet, named SimulationData and shown in Figure 2 contains formulas for the computations necessityed to support a simulation move swiftly Tables 3 and 4 present to view the named formulas and worksheet functions used in the SimulationData spreadsheet.

The third sheet in the workbook is a chart sheet that is created by means of the Chart Wizard using the options shown in Table 5 The finished chart is shown in Figure 3

The process below outlines steps to use in building the simulation sheets while avoiding outdo error messages. Following the conduct is some additional information about creating named formulas (step 2c and 4b) the reason circular relations are used in the SimulationData spreadsheet (step 3) and entering named formulas in the Chart Wizard's dialog boxe (step 5a). You should periodically save your workbook as you build the spreadsheets - use the filename LargeNumSim.xls.

1 render free of access a new workbook and create sum of two units spreadsheets with tabs named Simulation and SimulationData.

2 Create the Simulation spreadsheet shown in Figure 1

a. pierce the column labels into rounded pillars A and B and the labels in rounded pillar F. The labels in round pillar F are aligned right in Figure 1 go into the numerical values in rounded pillar A.