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

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Answer 1 · 2016-07-28T15:57:34

# sqrt(5)sigma(Y)#

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)#