What is the standard deviation of is sigma(X)=2sigma(Y), what is sigma(X+Y)?

1 Answer
Jul 28, 2016

sqrt(5)sigma(Y)

Explanation:

First note that
sigma(Z)=sqrt(Var(Z))
Var(Z_1+Z_2) = Var(Z_1)+Var(Z_2)
Var[a*Z]=a^2*Var[Z]
because

Var[(aZ-amu)^2] = Var[a^2(Z-mu)^2] = a^2Var[(Z-mu)^2]

using expected variance we obtain

sqrt(Var(X)) = 2*sqrt(Var(Y))

sqrt(Var(X)) = sqrt(4Var(Y))

sqrt(Var(X)) = sqrt(Var(2Y))

thus

sigma(X) = sigma(2Y)

this means every time we see sigma(X) we can replace it with
sigma(2Y)

sigma(X+Y) = sqrt(Var(2Y)+Var(Y))

sqrt(Var(2Y)+Var(Y)) =sqrt(4Var(Y)+Var(Y))=sqrt(5)sigma(Y)