Finding the new mean is simple: If [tex]E[X]=10[/tex], then [tex]E[2X]=2E[X]=20[/tex].
Meanwhile, the standard deviation is the square root of the variance, which is given by
[tex]V[X]=E[X^2]-E[X]^2\implies V[2X]=E[(2X)^2]-E[2X]^2=4E[X^2]-(2E[X])^2[/tex]
[tex]\implies V[2X]=4\underbrace{\left(E[X^2]-E[X]^2\right)}_{V[X]}=12[/tex]
and so the new standard deviation would be [tex]\sqrt{V[2X]}=\sqrt{12}=2\sqrt3[/tex].
