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?

Question

asked Apr 3, 2020 201k views

1 Answer

Casanova · Answer 1 · 2020-04-07T18:31:15+0000

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
$√(V[2X])=√(12)=2\sqrt3$ .