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Exercise 2: Illustrate the mean and variance of uniformly distributed random numbers.

For uniformly distributed real random numbers, , on [0,1), the average value should be 0.5. Note that, to compute the mean and variance, we can order the random numbers in any fashion (as both of these computations are independent of order). It is useful to choose to order the real, random numbers in ascending order, and approximate the distribution in the form of a continuous variable R. Verify by direct integration that the average, , and the variance, , should be 0.5 and 0.083333, respectively, as follows: