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Time Series Student Project Session: Fall 2011 Name: xxx xxx Time Series on Lottery Numbers Introduction Lottery is a national-approved gambling prevailed all around the world. In China, a welfare lottery called “Two Color Balls” is very popular among people. There are 33 red balls and 16 blue balls. The rule is to choose 6 out of the 33 red balls and 1 out of the 16 blue balls. This project focuses on the time series of the sum of the 6 chosen red ball numbers. Data Source All the data (100 values) are collected from (A Chinese webpage, if you are capable of Chinese language): http://www.cqcp.net/trend/ssq/trendchart_red.aspx A part of the data is as the following: Time Sum 2011109 77 2011110 134 2011111 81 2011112 89 2011113 117 2011114 108 2011115 86 2011116 75 2011117 121 2011118 77 Analysis 1) Stationarity From Figure 1, it is clear that this time series has no obvious trend. It is reasonable to assume stationarity in this time series. 2) Sample Autocorrelation Figure 2 displays the sample autocorrelation function from lag 1 to lag 99. All the points in Figure 2 are within the critical bounds 0.2 2 100 . 3) Partial Sample Autocorrelation Figure 3 is the graph of partial sample autocorrelation function from lag 1 to lag 99, which also confines all the points to critical bounds, the same as above. Figure 1 160 140 Sum 120 100 80 60 40 0 10 20 30 40 50 Time 60 70 80 90 100 Figure 2 0.2 0.15 0.1 Autocorrelation 0.05 0 -0.05 -0.1 -0.15 -0.2 0 10 20 30 40 50 Lag 60 70 80 90 100 60 70 80 90 100 Figure 3 0.15 0.1 Partial Autocorrelation 0.05 0 -0.05 -0.1 -0.15 -0.2 0 10 20 30 40 50 Lag 4) Y(t) versus Y(t-1) Figure 4 shows that there is no apparent upward or downward trend. Figure 4 160 140 Y(t) 120 100 80 60 40 40 60 80 100 Y(t-1) 120 140 160 5) Q-Q Plot After some simple calculations, I obtained that the mean of the sum is 101.47 and the standard deviation is 21.00. The Q-Q plot of figure 5 strongly suggests that Y t 101.47 is 21 approximately distributed as standard normal distribution. Figure 5 3 Quantiles of Input Sample 2 1 0 -1 -2 -3 -3 -2 -1 0 Standard Normal Quantiles 1 2 3 Conclusion From the previous analysis, the sum of red ball numbers is a white noise with mean 101.47 and standard deviation 21, Y t 101.47 t , where t is a random error with mean 0 and standard deviation 21. Figure 1 shows no need to difference or log the original data. Figure 2 and 3 show that any MA(p) or AR(q) is inappropriate. Figure 4 reveals no direct connection between Y(t) and Y(t-1), which leads to consider white noise process. Figure 5 finally confirms the hypothesis. Comment on conclusion It is no surprise that the sum of red ball numbers is a white noise process. Predicting lottery numbers is really challenging. But one thing for sure should be bore in mind: keep the sum within one standard deviation away from the mean, since events like extremely large or small sums are unlikely to occur.