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
`√(V[2X])=√(12)=2\sqrt3`.