A basketball player practiced free throws for 200 shots. The observed outcomes were: Made (146 shots) and Missed (54 shots). Define a scoring variable as follows: for a made free throw and for a missed free throw. Using the relative frequencies from these 200 shots as estimated probabilities, what is the expected value of based on the empirical distribution?
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Explanation: Empirical expected values involve estimating probabilities from frequencies and computing a weighted average of outcomes. For 200 free throws, made (2 points) 146 times and missed (0 points) 54 times. Estimated probabilities: 146/200 for 2, 54/200 for 0. E[X] = 2*(146/200) + 0*(54/200) = 292/200 = 1.46. This is the average points per shot from the data. The approach emphasizes empirical estimation, where the expected value is the sample mean, a weighted average that approximates the true value and may vary across samples.