Читать книгу Informatics and Machine Learning. From Martingales to Metaheuristics онлайн

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7 2.7 Prove that relative entropy is always positive (hint: use Jensen's Inequality from ssss1).

8 2.8 What is the Expectation for the two‐dice roll with pair outcome probabilities listed in (Exercise 2.3)?

9 2.9 What is the Expectation for the two‐dice roll with fair dice? Is this expectation an actual outcome possibility? What does it mean if it is not?

10 2.10 Survey the literature and write a report on common occurrences of distributions of the type: uniform, geometric, exponential, Gaussian, log‐normal, heavy‐tail.

11 2.11 Survey the literature and write a report on common occurrences of series of the type: Martingale.

12 2.12 Consider the loaded die example, where the probability of rolling a 1,2,3,4, or 5, is 0.1, and the probability of rolling a 6 is 0.5.What is the expectation for the loaded die?What is its variance?What is its mode?What is its median?The LLN for the loaded die above indicates that a sequence of rolls could be done and if its average tends toward 4.5, you know it is loaded, and if it goes toward 3.5, you know it is fair. So it comes down to how soon you can resolve that its converging on these two possible expectations differing by 1.0. Suppose someone is rolling a die that is either fair or loaded as described above, how many rolls do you think you will need to see before it will be obvious how the average is trending? Is the better way to spot the anomaly? Like frequency of seeing three sixes in a row is notably skewed?