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Consider independent random variables, $X_1, ..., X_n$ with mean $\mu$ and variance $\sigma^2$

Let $Z = \sum_{i=1}^{n} X_i$

Then what is the approximate normal distribution of $Z + 6$?

Clearly $Z \sim N(n\mu, n\sigma^2)$

but when I do $Z + 6$, what happens to the distribution?

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    $\begingroup$ Adding a constant to a random variable does not change the variance, but it adds the same constant to the expected value. Adding a constant to a normally distributed random variable yields a normally distributed random variable. $\endgroup$ – Michael Hardy Apr 18 '17 at 5:12
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If $Z \sim N(n \mu, n \sigma^2)$, then $Z+6 \sim N(n\mu +6, \sigma^2)$.

Note that the mean is translated accordingly but the variance is not affected.

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