Expectation calculation

[elhazzag]

Hi,

I have an exercise on which I am struggling:

Suppose X is a Gaussian variable of mean 0, and standard deviation σ,
and define N(u) as the CDF of a centered Gaussian variable of mean 0 and
variance 1,
Calculate the expectation of N(X).

Can someone please help or give guidance on this?

 

[Keith Tan]

is it 0.5?

 

[elhazzag]

Is it because N(X) will follow a uniform law?

 

[jiakewei]

the mean is zero or the letter o?

 

[elhazzag]

The mean is zero

 

[jiakewei]

The mean is zero

I think it is 1/2 times one over square root 2*pi.

 

[elhazzag]

Not this is correct. How did you get this result?
I have run a simulation in excel and I get 0.5 for whatever standard deviation I chose for X.

 

[amraj]

Were you running your simulations for sigma=1? In that case, the density in question can easily be shown to be uniformly distributed on (0,1) by considering the CDF of N(X). I'm pretty sure the same approach can be used for a general sigma.

 

[QTMGMT]

You can prove that N(x) has a uniform distribution on [0,1]. Then you can use properties of the uniform distribution to calculate the mean. which you will find is 0.5

 

[elhazzag]

How would you prove that N(X) follows a Uniform distribution?

 

[QTMGMT]

first use this:
http://www.seas.gwu.edu/~dorpjr/EMSE 001/Inverse CDF Theorem.pdf

then use this:
http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

 

[C S]

Nothing wrong with getting a little fancy like in the given answers, but you can very simply note the symmetry of the normal distribution (N(-x) = 1 - N(x)). Use this when setting up the integral for the expectation of the cdf to write it as three integrals. You'll note a cancellation which leaves you integrating the pdf of the normal but only over "half" the real line.