Linear transformation of a random variable

For $Y = a + bX$, the mean moves exactly the way the values move, $\mu_Y = a + b\mu_X$ (mu, the mean), but the standard deviation responds only to the multiplier: $\sigma_Y = |b|\sigma_X$ (sigma, the standard deviation). Adding a constant slides the whole distribution without stretching it, so it leaves the spread untouched, and the absolute value keeps the standard deviation positive when $b$ is negative. Suppose $X$ has $\mu_X = 20$ and $\sigma_X = 4$. Then $X + 7$ has mean $27$ with standard deviation still $4$, while $100 - 3X$ has mean $100 - 3(20) = 40$ and standard deviation $|-3|(4) = 12$.