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4.5 Expectation

Expectation of a Uniform Distribution

Let X be a random variable with uniform distribution over the integers from -n to n. Let Y := X^2.

Which of the following are true?

Exercise 1

  1. Since PDF_X(x) is symmetric around 0, we know the mean has to be 0.
  2. All values of Y are nonnegative and most of them are actually positive with non-zero probability. There is no way for the mean to be 0. It has to be some positive value.
  3. Obvious, since E[X]=0 and E[Y]>0.
  4. True by linearity of expectation, no matter what X and Y are.
  5. This is tricky. The equation does not hold in general for non-independent X and Y. However, in this particular case, it happens to hold. To see this, note that the right hand side is 0, since E[X]=0. At the same time, the random variable XY=X^3. Its PDF is symmetric around 0, so its mean must be 0 as well.
  6. X and Y are obviously not independent.
  7. E[Y]=\sum_{i=0}^n 2i^2\frac{1}{2n+1} < \sum_{i=0}^n 2i^4\frac{1}{2n+1} = E[Y^2].