Question

A random variable X has a mean of 10 and a standard deviation of 3. If each value of X is multiplied by 2, what will the new mean and standard deviation be?

Answer 1

Finding the new mean is simple: If $E[X]=10$, then $E[2X]=2E[X]=20$.

Meanwhile, the standard deviation is the square root of the variance, which is given by

$V[X]=E[X^2]-E[X]^2\implies V[2X]=E[(2X)^2]-E[2X]^2=4E[X^2]-(2E[X])^2$

$\implies V[2X]=4\underbrace{\left(E[X^2]-E[X]^2\right)}_{V[X]}=12$

and so the new standard deviation would be $\sqrt{V[2X]}=\sqrt{12}=2\sqrt3$.