# Math Help - How many averages are enough?

1. ## How many averages are enough?

Hi,

I have been trying to figure this out for two days but I don't seem to be reaching a conclusion

Let me describe what I'm doing...

I have a vector of constants: $X$
I obtain a vector of random values, $Z$ by adding a vector of AGWN, $N$, to $X$: $Z=X+N$

Finally, I compute $C$ between $X$ and $Y$: $C=\sigma_{XZ}/\sqrt(\sigma_{X}\sigma_{Z})$

As you can see, the value of $C$ would change each time I generate a new vector of AWGN noise. Therefore I will average $C$ over several trials.

What I can't get my head round is how can I decide how many trials are enough?

I have gone on various errands to standard error of the mean, confidence intervals, and Monte Carlo trials, without fruition.

Thanks.

2. ## Re: How many averages are enough?

You seem to have neglected to tell us what you are trying to do!

3. ## Re: How many averages are enough?

Oh yes...here goes

I want to obtain an accurate value of $C$. I know that sounds very vague - what do I mean by "accurate"?...

Well, I could go on taking averages of $C$ ad infinitum to get a better and better estimate of $C$. This isn't practical, and also I can't just decide to take 10 000 averages because it looks big enough.

What I must do is determine how many averages of $C$ give a 'good' representation of the r.v. $C$.

My knowns are:

$X:$ A vector of constants

$Z = X + AWGN$

$AWGN$ is white noise of $N(\mu, \sigma)$

and $C$ is defined as $C = \sigma_{XZ}/\sqrt(\sigma_{X}\sigma_{Z})$. The value of $C$ will fluctuate as $AWGN$ (white noise) is different each trial.

Thanks.

4. ## Re: How many averages are enough?

I hope my description hasn't obscured things, it is a simple concept I'm sure. Please tell me if anything needs any clarification.

Let me try and sum it up very simply:

All vectors have the same length.

$Step 1. Z = X$ (vector of known constants) $+ White Gaussian Noise$ (known variance and mean)
$Step 2. C = \sigma_{XZ}/\sqrt(\sigma_{X}\sigma_{Z})$ (i.e. the correlation coefficient between $X$ and $Z$).

My objective: Repeat Steps 1 and 2 $N$ times to obtain a simple average of $C$. The question: what approaches can I take for deciding $N$?

Thanks.