The distribution of a sum of independent normal variables is which of the following?

• A. Normal(sum mu, sum variances)
• B. Normal(sum mu, product of variances)
• C. Normal(total mu, sum variances)
• D. Normal(mu, sum variances)

Answer: (A)

When you add independent normal variables, the result is still normal. The mean adds, and the variance adds as well, provided the variables are independent. So the sum has distribution Normal with mean equal to the sum of the individual means and variance equal to the sum of the individual variances. For example, if X1 ~ N(mu1, sigma1^2) and X2 ~ N(mu2, sigma2^2) are independent, then X1 + X2 ~ N(mu1 + mu2, sigma1^2 + sigma2^2). This extends to any number of variables.