Stretch and move normal distribution while keeping the sum 1

Question

I'm currently trying to distribute 100% (1) over 20 points.
So far the standardized gaussian normal distribution did some good work with a standard deviation of 2 and a mean of 10, which distributes roughly 1 (0.999..) over my 20 points: Looks like this.

Now, I can just increase the deviation a lot but that doesn't add up to 1 anymore and stops being an equal distribution: 1 is bigger than 20 and the sum is 0.9~.

Is there a way to stretch and move the normal distribution while keeping the sum at 1?

Answer 1

• Translation of mean location does nothing to the sum.
• For the variance we need to scale as follows $\sigma^2 \mathcal{N}(0,t) =\mathcal{N}(0,\sigma^2 t)$

What does the last statement mean? Well, the normal distribution if multiplied by a scalar will automatically adjust the variance to yield a normal distribution so the sum is maintained (this is pretty nice property of a normal distribution)

From the comments it seems you want a truncated normal distribution

So if you want a normal distribution that sums to 1 between a predefined interval then you can use $$ f(x; \mu, \sigma, a, b) = \frac{\frac{1}{\sigma}N\left(\frac{x-\mu}{\sigma}\right)}{\Phi\left(\frac{b-\mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)} $$ where the interval is $-\infty \leq a < b \leq \infty$ and $\Phi(x)$ is the cumulative normal distribution.