# What is the maximum value of a measured random variable

This is a statistics question but I don't know how to phrase it in statistician's parlance, so I'm going to ask it in electrical engineering terms. You should be able to follow it easily though. I'm hoping that it won't be migrated /closed as it really is just applied statistics.

I have a totally random voltage that is called white noise. It's the stuff you hear in between radio stations on your tranny. That means that it is effectively normally distributed. By the definition of white noise, there is no simple upper limit to the rate at which this noise can change it's value. It may be millions of times a second. I don't know the mean or standard deviation, but when I look at a graph of it's values, they kinda go from 0 - 1 volt. Some times due to the random nature, they go higher. They might reach 1.5 volts. This is all assessed by inspection alone. I have no control over this noise as it is generated by physical processes and quantum mechanics.

I now sample this random voltage 10,000 times during the period of 1 second. With these samples recorded, is it possible to say with any confidence what the maximum voltage might be? And what would that confidence be?

This question is reminiscent of my A Level maths statistics but I'm having trouble applying it. I believe that it should be possible to determine something like it is <1.8 volts for 99% of the time. I'm really looking for concrete numbers rather than a theorem as I need to build stuff to utilise these findings.

With 10,000 samples, the simple sample maximum should be quite a good estimate of the population maximum. However, if the variable in question is normally distributed, that means it isn't bounded above, so there is no maximum.

I believe that it should be possible to determine something like it is <1.8 volts for 99% of the time

That would be the 99th percentile. Again, the sample 99th percentile is likely a good estimator in this situation. To find the sample 99th percentile, sort all your data points and then pick the value that's 99% of the way to the greatest (e.g., if you have 10,000 data points, pick the 9,900th).

• What would I know by taking the highest value of the 10,000 samples? Would that be useful? My sampling machine can only store about 150 samples unfortunately. – Paul Uszak Aug 20 '16 at 0:35
• So would the highest value be 99.99% confident then? Is it that easy? – Paul Uszak Aug 20 '16 at 2:52
• Well, the trick is that more extreme quantiles are harder to estimate accurately. With a sample size of 10,000, you can likely estimate the .99 quantile well, but the .9999 quantile only inaccurately. You can get a sense of how accurate your estimate is by constructing a confidence interval. – Kodiologist Aug 20 '16 at 3:17
• The sample maximum of your 10,000 points is a good "soft upper bound" for the random variable, even if it isn't a particularly good estimate of the .9999 quantile. – Kodiologist Aug 20 '16 at 3:19
I now sample this random voltage 10,000 times during the period of 1 second.
With these samples recorded, is it possible to say with any confidence what
the maximum voltage might be?


You have a sequence $\{X_1,\ldots,X_n\}$ of i.i.d. random variable with $X_i\sim N(\mu,\sigma^2)$. If I understood your question right, you are concerned with statistical properties of:

$$Y_n = \max\{X_1,\ldots,X_n\}$$

If the cdf of the $X_i$ is differentiable (which is true when the density of $X_i$ is normal) we have:

$$f_{Y_n}(y) = nF_X(y)^{n-1}f_X(y),$$

where I'm using $F_X$ and $f_X$ to denote the common cdf and density function of the $X_i$. In your particular situation you have $f_X$ as a $N(\mu,\sigma^2)$ and $F_X$ is the cumulative function of this density.

I believe that it should be possible to determine something like it is <1.8 volts for 99% of the time


To find $y_{\alpha}$ such that $\mathbb P(Y_n \leq y_{\alpha}) = \alpha$ you need to solve:

$$\int_{-\infty}^{y_{\alpha}}f_{Y_n}(y)dy = \int_{-\infty}^{y_{\alpha}}nF_X(y)^{n-1}f_X(y)dy = \alpha$$

If no analytical solution exist a numerical approximation would do the job.

For the method just discussed you need values of $\mu$ and $\sigma^2$. If you don't have them at hand estimating them with:

$$\hat{\mu} = \frac{1}{n}\sum_{i=1}^nX_i$$ $$\hat{\sigma^2} = \frac{1}{n-1}\sum_{i=1}^n(X_i-\hat{\mu})^2$$

is usually the best option.

• I think OP was asking about the population maximum, not the expected maximum of a fixed-size sample. – Kodiologist Aug 20 '16 at 3:34
• The question states "That means that it is effectively normally distributed". Under this assumption there is no meaningful way of talking about a population maximum. – Mur1lo Aug 20 '16 at 3:59
• @Mur1lo I suspect that this is the distinction between theory and practice. There may be no theoretical maximum, but a circuit has to be built out of real components that doesn't miss the peak values. Or at least a lot of the time doesn't. I could just build for a 100V infrequent peak to be super safe, but (respectfully) I was hoping for something cleverer. – Paul Uszak Aug 20 '16 at 4:12
• @PaulUszak In my answer I tried to address the issue as you formulated it in your question. From the formulation alone I was unable to predict the practical purpose you are aiming at. – Mur1lo Aug 20 '16 at 4:54