How to Represent a Gaussian Distribution in 1D?

[dacruick]

Hi,

I currently have a Gaussian distribution (Normalized Frequency on the y-axis and a value we can just call x on the x-axis).

So for the sake of simplicity, let's say that I ignore any values below 0 and any values above 1 on the x-axis. Then what I will do is take 10 equal segments (0.1 units in length) of the remaining gaussian distribution. Then this is where I get confused. What I want to have is a 1 dimensional representation of this distribution. The y-axis will be represented by the distances between two x-values. So for example, if the slope between 2 points is a large positive number, the distance between the two points will be decreasing because the frequency is increasing. If the slope is 0, then the distance between the points will be constant, because there is a constant frequency.

Does this make any sense?

I feel like this idea can be easily done and the solution is dangling right in front of me but I just can't seem to get it.

Thank you in advance for any help.

dacruick

 

[mmwave]

shank

Hi dacruickshank,

Thank you for sharing your idea and question with us. It sounds like you are trying to find a way to represent your Gaussian distribution in a 1-dimensional form. This can definitely be done and there are a few different approaches you can take.

One approach is to use the cumulative distribution function (CDF) of your Gaussian distribution. This function takes into account the entire distribution and maps it onto a 1-dimensional scale, with values ranging from 0 to 1. The CDF essentially shows the probability that a random variable will take on a value less than or equal to a given value. So, in your case, the x-axis would represent the different values of x and the y-axis would represent the cumulative probability.

Another approach is to use the first derivative of your Gaussian distribution. This would give you a measure of the slope at each point on the distribution, which you could then use to calculate the distances between points as you described in your post. This approach may require some additional mathematical calculations, but it could give you the 1-dimensional representation you are looking for.

I hope this helps and gives you some ideas to explore. Good luck with your research!