The sample data x1, x2, …….., xn sometimes represents a time series, where xt = the obse...

The sample data x₁, x₂, …….., x_n sometimes represents a time series, where x_t = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longerterm behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with