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FRM Part I · FRM Exam Part I · Multivariate Random Variables

Returns X and Y on two equity funds follow a bivariate normal distribution with correlation 0.60. Which statement about the relationship between X and Y is correct?

Under a bivariate normal distribution, Y conditional on X = x is normal with a mean that is a linear function of x and a constant variance. Marginals are always normal, and zero correlation implies independence, so the other statements are false.

  1. AConditional on X = x, Y is normally distributed with a mean that is linear in xCorrect
  2. BZero correlation between X and Y implies they are independent only if both marginals are uniform
  3. CThe conditional variance of Y given X = x increases as x moves further from the mean of X
  4. DThe marginal distribution of Y is normal only if the correlation equals zero

Explanation

For a bivariate normal, the conditional distribution of Y given X = x is normal with mean mu_Y + rho(sigma_Y/sigma_X)(x - mu_X), which is linear in x. The conditional variance is constant, sigma_Y^2(1 - rho^2), so the third option is wrong. Marginals are always normal, and zero correlation implies independence in the bivariate normal case.

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