Hello,
If a and d are indeed independent, then your working is correct
I have values a,b and d.
a and d have distribution N(0,1)
b has distribution N(0,1/2)+i*N(0,1/2)
notation - b* is complex conjugate of b
I need to evaluate the following;
E[a^2 + d^2 - 2ad + 4bb*]
= E[a^2] + E[d^2] -2*E(ad) + 4E[bb*]
At the moment I have the answer as 6, but running the numerics keeps giving me the answer of 4. I'm now starting to doubt my answer, so please can you check whether it is correct for me?
My working
using the variance formula I have
var(a) = E[a^2] - E[a]^2
so E[a^2] = var(a) + E[a]^2 = 1 + 0 = 1
so E[d^2] = 1
Now, the values of a and d have been chosen at random under the distribution => E[ad]=E[a]E[d]= 0
The next part is where I feel it may be going slightly wrong.
E[bb*]= E[ Re{b}^2 + Im{b}^2] = E[Re{b}^2] + E[Im{b}^2]
using the variance formula from earlier.
E[Re{b}^2] = var[Re{b}] + E[Re{b}]^2 = 1/2
E[Im{b}^2] = var[Im{b}] + E[Im{b}]^2 = 1/2
so E[bb*] = 1
then
E[a^2 + d^2 - 2ad + 4bb*] = 1 + 1 - 0 + 4 = 6.
Is this all correct?