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In the exercises for the MAT8181 course, there are two Exercises (16 and 17) about prediction and extrapolation based on MA(1) and AR(1) time series. But before discussing those exercises (I had some request for hints), I wanted to recall the definition of the linear prediction,whereAs discussed previously on this blog, we consider here on projection not on  (this would be the conditional expectancy) but on the linear subset.The goal of Exercise 2 was to establish an important result, in the context of Gaussian random vectors. If  is a (multivariate) Gaussian vector, , thenwhere  is the vector .Keep those results in mind, and let us look at Exercise 17, for instance. Here,  is an AR(1) process, with innovation ,One observation (say ) is missing. We have here 3 questions:what is the best linear prediction of  given  and what is the best linear prediction of  given  and what is the best linear prediction of  given  and Case 1. Here, we have to computeSince we have an AR(1) process, ,  and . Thus, from the relationship abovewhich can be writteni.e. . Which makes sense actually: the AR(1) process is Markovian of order one, soAnd we seen in class that for an AR(1) processSo far, so good.Case 2. Here, we have to computeSince we have an AR(1) process, ,  and . Thus, from the relationship abovei.e. .Case 3. Finally, we have to computeOne more time, since we have an AR(1) process, ,  and . So here, the relationship above becomesHere, we can writei.e.So finally, what we got here isandThe mean squared errors for each of those estimates are obtained computingI guess I should probably stop here... that's a detailed hint actually.