Statistics - proof. Please help?
Prove mathematically what happens with the sample variance (one of the measures pf the variability) when multiplying all values of the data by a nonnegative constant b.
- MichaelLv 74 years agoFavorite Answer
edit : thanks for Randy's answer, nevertheless he is not correct
Sum [ (x_i)^2 ] = sum [ b^2 x_i^2 = b^2 sum x_i^2.
VarX = Sum [ (x_i)^2 ]
which is only correct if E(X) = 0
first of all, welcome to this forum from (maybe the only) french guy in the zone !! ;-)
by definition the variance of a RV X is :
VarX = E( [ X - E(X) ]^2 <------------------ E(X) is the mean of X
as the E( . ) is linear :
it can easily been proven that, after expansion we obtain the theorem :
VarX = E(X^2) - E(X)^2
now let's examine Y=bX then,
VarY = E( [bX]^2) - E( [bX] )^2
= E( b^2 X^2 ) - E( bX )^2
= b^2 E(X^2) - [ E(bX) ]^2
= b^2 E(X^2) - [b* E(X) ]^2
= b^2 E(X^2) - b^2* E(X)^2
= b^2 (E(X^2) - E(X)^2 )
= b^2 VarX
Var(bX) = b^2 VarX
comment : the condition b >= 0 is not necessary
hope it' ll help !!
PS: if you want good answers, please do not forget to give Best Answers to one of those who answer.....
to me, or anybody else, or course provided the answer deserves one !
this is to encourage people to answer !! ;-)
- Randy PLv 74 years ago
Apply the definition of variance.
Sum(b*x_i) = b sum(x_i).
Sum [ (b*x_i)^2 ] = sum [ b^2 x_i^2 = b^2 sum x_i^2.
You're in an advanced math course, you shouldn't need help writing down those facts. Now write down the expression for variance of bX and apply those facts.