The value of E( (||X-μ||-c)^2 )

[saintman4]

I would like to ask one more question.

X ~ N (μ, σ2)

If X = [x1 x2] and μ = [μ1 μ2]. What is the value of E( (||X- μ||-c)^2 )?

where c is constant and E(||X- μ||^2)= σ2


Thank ...

 

[Pere Callahan]

$$\mathbb{E}[ (||X- \mu||-c)^2 ]=\mathbb{E}[ ||X- \mu||^2-2c||X-\mu||+c^2 ]$$

Now $\mathbb{E}[\cdot]$ is linear ... does this help?

 

[saintman4]

Thank Pere Callahan. But I still don't know how to find E(||X- μ||) as I ask in https://www.physicsforums.com/showthread.php?t=224947

Do you know how to find E(||X- μ||)?

Thank you.

 

[Pere Callahan]

Did yu try to use the definition of the expected value of some function g of a real random Variable X with density function f?
$$\mathbb{E}[g(X)]=\int_{\mathbb{R}}{g(x)f(x)dx}$$

You will have to solve some Gaussian-type integrals but it should be straight-forward