Random Sampling

In the real world, most R.V.’s for practical applications are continuous, and have no generalized formula for f X (x) and F X (x). We may approximate the density functions by taking a random sample, with a large enough sample size, n, and plot the relative frequencies within the sample.

Random Sampling Examples:  Suppose you wanted to know more information about the GPAs of students enrolled at the U of A  Rather than look up every individual student, you can take a small sample of randomly selected students and figure out their GPAs to project what the GPAs of the entire student body would be.  Taking a poll of registered voters for the presidential election  Looking at small sample of closing price for a particular stock

Random Sampling The whole idea behind random sampling is to let a part represent the whole.

Random Sampling—An Example Suppose that X is the number of assembly line stoppages that occur during an 8-hour shift in our manufacturing plant. We could obtain a random sample of size 10 by watching the line for 10 different shifts and recording the number of stoppages during each eight hour shift.

Example (cont) The table below shows the number of work stoppages for various shifts: A histogram plot of these stoppages gives us a pictorial representation of how this chaotic data behaves.

Example (cont) The relative frequency histogram plot is shown below:

Example (cont) Our histogram plot of the relative frequencies for the work stoppage example can be used to approximate the p.m.f. for this situation Of course, if we increase the number of observations of shifts, our p.m.f. will be more accurate.

Example (cont) From our example, we can also look at the average number of work stoppages: The average we just found is sometimes called the sample mean and can be found using:

Example 2 Suppose that the assembly line discussed in Example 1 runs 24 hours per day, with workers in three shifts. The sheet Numbers in the Excel file Stoppages.xls contains records of the number of stoppages per shift for nine months (819 shifts).Example 1

Example 2 Computations in that sheet show that the number of stoppages in the sample ranged from 0 to 14, with a mean of The sample in Stoppages.xls is much larger than the one of size 10 that we considered in the previous example. Hence, we would replace the earlier estimate of 6.3 for E(X) with the new estimate of 5.78.

Example 2 A histogram plot of the relative frequencies also give us a good estimation for the p.m.f.

Example 3 We can also use a large sampling to approximate the p.d.f. for a continuous random variable. The manager of the plant that we discussed in previous examples wants to get a better understanding of the delays caused by stoppages of the assemble line. In addition to the number of stoppages she is also interested in how long they last. Let T be the length of time, in minutes, that a randomly selected stoppage will last. The duration of each of the 4,734 stoppages that occurred during the 819 shifts was recorded. This provides a random sample of observations of the continuous random variable T. The times are shown in the sheet Times in the Excel file Stoppages.xls.

Example 3 (cont) The histogram of times is converted to relative frequencies. We would like to treat this as a p.d.f.  This means the total area must be 1!  To do this the area of each rectangle of our histogram must equal the relative frequency.  Because we already have made our bins of width 2 we must adjust the heights of our relative frequency so that the area of each rectangle equals the relative frequency.  This is done by taking the relative frequencies and dividing by the bin width  Notice for example the bar whose bin label is 7 has a height of 0.07, the area of this bar = 2 * 0.07 = 0.14, which is the relative frequency for this bin.

Example 3 (cont) A histogram plot gives us an approximation for the p.d.f. of our random variable T

Example 3 (cont) The graph of the actual p.d.f. f T is a smooth curve. We can approximate this curve by connecting the midpoints at the tops of the bars in our histogram. The resulting piecewise linear graph is show below, and in the sheet Times of Stoppages.xls

Class Project 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.  Visually, what do the histograms tell you about the volatilities of your stock? How does E(R norm ) compare to the weekly risk- free ratio, R rf ? Why does this happen?