Sums of normals

  • Consider a normal random variable \(X\) with mean \(\mu_1\) and standard deviation \(\sigma_1\)
  • Consider a normal random variable \(Y\) with mean \(\mu_2\) and standard deviation \(\sigma_2\).

Assume that \(X\) and \(Y\) are independent and define \(Z=X+Y\)

  1. What is the distribution of \(Z\) ?
  2. What is the mean and variance of \(Z\) ?
  3. (**) If we now assume that they are not independent, but still normal as described above, what can you say ?

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