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Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$. Then the distribution of $X$ is

  1. Gaussian random variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$.
  2. Uniform with mean $\mu\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)$.
  3. Gaussian random variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$.
  4. Has no known form.
  5. None of the above.
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