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Math and science::INF ML AI
Sum of Gaussian random variables
Sum of Gaussian random variables
Let \( X : \Omega \to \mathbb{R} \) and \( Y : \Omega \to \mathbb{R} \) be
two random variables each with a Gaussian distribution:
- \( X \sim \mathcal{N}(\mu_x, \sigma_x^2) \)
- \( Y \sim \mathcal{N}(\mu_y, \sigma_y^2) \)
Then the random variable \( Z = X + Y \) has a Gaussian distribution:
\( Z \sim \mathcal{N}(\mu_x + \mu_y, \sigma_x^2 + \sigma_y^2) \)
Follow-up. \( Z = XY \) [does/n't] have a Gaussian distribution?
Can you remember the geometric intuition/proof?
