Thread: Finding E[|b|^4] = E[(bb*)^2]

I need to work out E((bb*)^2) for b~N(0,1/2)+iN(0,1/2) b* is complex conjugate.

Using prior working:
E[bb*]= E[ Re{b}^2 + Im{b}^2] = E[Re{b}^2] + E[Im{b}^2]
using the variance formula
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

So far I have that:

E[(bb*)^2] = E[Re{b}^4 + Im{b}^4 + 2*Re{b}^2*Im{b}^2]
= E[Re{b}^4] + E[Im{b}^4] + 2*E[Re{b}^2]*E[Im{b}^2]

which tells me that 2*E[Re{b}^2]*E[Im{b}^2] = 2*1/2*1/2 = 1/2
But I am unsure of how to compute E[Re{b}^4] and E[Im{b}^4] easily as I do not know the variance of bb*

Please can someone verify my working so far, show me how to compute the rest of the expectation and tell me what answer I should arrive at (so I can check working etc)?

Thankyou

Originally Posted by ash89

I need to work out E((bb*)^2) for b~N(0,1/2)+iN(0,1/2) b* is complex conjugate.

Using prior working:
E[bb*]= E[ Re{b}^2 + Im{b}^2] = E[Re{b}^2] + E[Im{b}^2]
using the variance formula
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

So far I have that:

E[(bb*)^2] = E[Re{b}^4 + Im{b}^4 + 2*Re{b}^2*Im{b}^2]
= E[Re{b}^4] + E[Im{b}^4] + 2*E[Re{b}^2]*E[Im{b}^2]

which tells me that 2*E[Re{b}^2]*E[Im{b}^2] = 2*1/2*1/2 = 1/2
But I am unsure of how to compute E[Re{b}^4] and E[Im{b}^4] easily as I do not know the variance of bb*

Your work seems correct so far. What you would need is when , i.e. compute

You can integrate by parts by writing the integrand as , deriving the first factor and integrating the second one. Thus you will resume to something like for some constant C, and you can relate this to

.

You should find , I think. Another way consists in differentiating the characteristic (or moment generating) function four times and evaluating it at 0, if you know that technique.