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IAI Actuarial Core Principles · Actuarial Statistics · Jointly distributed random variables

X is standard normal and Y = X². Which statement about Corr(X, Y) is correct?

The correlation is 0 but X and Y are dependent. Cov(X, X²) equals E[X³] minus E[X]E[X²], and both terms are zero by symmetry of the standard normal. Zero correlation only rules out linear association, not dependence.

  1. AIt equals 1 because Y is a function of X
  2. BIt equals 0, and X and Y are therefore independent
  3. CIt equals 0, although X and Y are dependentCorrect
  4. DIt equals -1 because Y is always non-negative
  5. It equals 0.5 as Y is quadratic

Explanation

Cov(X, X²) = E[X³] - E[X]E[X²] = 0 - 0 = 0 because the standard normal is symmetric. So the correlation is 0, yet Y is completely determined by X, so they are dependent. Zero correlation does not imply independence.

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