# Variance of X squared

#### gigiola

##### New Member
I have a simple question, but very tricky...
I have a random variable Z with:
E[Z]= S
var (Z)= sigma

what is var(Z^2)?
thanks

#### vinux

##### Dark Knight
If you know the distribution then you can calculate.
var(Z) =E[ (Z-E[Z])^2 ]
var(Z^2) =E[ (Z^2-E[Z^2])^2 ]

#### gigiola

##### New Member
I don't know the distribution (it is a random variable), and I have no value of this variable (so I can not calculate manually).

#### Dason

If you know the distribution then you can calculate.
var(Z) =E[ (Z^2-E[Z])^2 ]
var(Z^2) =E[ (Z^4-E[Z^2])^2 ]
vinux I think you mixed up a little bit. I know of two equivalent ways of getting variances but it looks like you combined them to get something wrong.

$$Var(Z^2) = E[(Z^2 - E[Z^2])^2] = E[Z^4] - E[Z^2]^2$$

#### vinux

##### Dark Knight
You are right Dason.. I was careless.

gigiola,
if distribution is not specified,
May be you can assume normal distribution and come up an answer of Var(Z^2)

E[Z^4] = S^4 + 6(S^2)σ + 3σ^2
E[Z^2] = σ + S^2
Now you can calculate Var(Z^2)
http://en.wikipedia.org/wiki/Normal_distribution

#### gigiola

##### New Member
on internet I found something like:
var(XY)=Var(X)var(Y)+var(X)E[Y]^2+var(Y)E[X]^2
but there is the assumption of Yand X independent, that is hardly my case.

Probably I am complicating too much the question, so I give you the equation I have to solve, because probably there is an easier solution:

E[-e^-alpha(Y+Z)^2] that is:
-e^-alpha[E[Y+Z]^2-alpha/2*var[(Y+Z)^2] with Y and Z distributed as normal

thank you

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