Let X be a continuous random variable. What value of b minimizes E (|X-b|)? Giv

[johnG2011]

Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Giv

Homework Statement

Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation

The Attempt at a Solution

E(|X - b|)

E[e - $\bar{x}$] = E(X)

E(|E[e - $\bar{x}$] - b|)

so ?,... 0 = E(|E[e - $\bar{x}$] - E|)

but this is a graduate course, I have a funny feeling that I am supposed to derive this using a the integral of an Expected value.

 

[lanedance]

what is e? Its not clear what your steps are attempting

I think the integral would be a good way to approach this

 

[johnG2011]

what is e? Its not clear what your steps are attempting

I think the integral would be a good way to approach this

The e is supposed to be an observation in the sample set

 

[lanedance]

ok well its still not real clear what you're trying to do

i would try and write the expectation in integral form and consider differentiating, though you may need to be careful with the absolute value
$$f(b) = E[|X-b|] = \int_{-\infty}^{\infty} dx.p(x).|x-b|$$

 

[lanedance]

if the absolute value sign gives you trouble, you could consider using b to break up the integral into a sum of two integrals (x<b and x>b), this however will complicate the differentiation as now b appears in the integration limit also