# Notation and Transformation Rules of the Normal Distribution

So, I'm wondering. So far I have seen both notations:
(1) $$X \sim \mathcal{N}(\mu, \sigma^2)$$ and
(2) $$X \sim \mathcal{N}(\mu, \sigma)$$ where $$\mu$$ is the mean. My first questions is directed to $$\sigma$$.

In (1), does it resemble the variance and in (2) the standard deviation? I assume there is no 'correct' notation, but it is rather a choice of preference, isn't it?

I also saw the following notation
(3) $$\mu + \sigma \cdot \mathcal{N}(0, 1)$$

Which of the aforementioned notations, i.e., (1) and (2), can be transformed to (3)? Or can both be transformed to (3) without violating any fundamental rules? And what does $$\sigma$$ in (3) refer to? Is it the standard deviation of $$\mathcal{N}$$?

• The confusion occurs not so much between $N(\mu, \sigma^2)$ and $N(\mu,\sigma)$, because that notation itself suggests what convention is followed, but by notation such as $N(\mu, 20)$. Is 20 the variance or sd? In my view, the overwhelming convention is that the second parameter denotes the variance ... but unfortunately, exceptions exist, and so the only rigrorous solution is to define one's notation. Commented Feb 10, 2019 at 15:15
• One reasonably unambiguous solution is to write $N(\mu,(\sqrt{20})^2)$. It's not as good as an explicit definition such as wolfies calls for, but it's often sufficient for people to understand that you're talking about $\sigma^2=20$ (otherwise why the rigmarole of taking square roots?) Commented Feb 11, 2019 at 1:04

The easiest way to understand that the choice of parameter results in equal outcome can be found in the probability density function (PDF). The PDF of the Normal distribution is given by:

$$\mathcal{N}(\mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{ -\frac{(x-\mu)^2}{2\sigma^2}}$$

The function is parametrized by $$\mu$$ (mean) and $$\sigma$$ (standard deviation) or $$\sigma^2$$ (variance). Both are equivalent, so is merely at the user's convenience and convention. For completeness I note that there is another parameterization people sometimes choose: its precision, given by $$\frac{1}{\sigma^2}$$.

In the third notation, you find the standard Normal distribution $$\mathcal{N}(0, 1)$$. The parameters are used to shift and scale the distribution. Here it does matter that we use the variance (as scaling by $$\sigma$$ multiplies the variance by $$\sigma^2$$)!

To illustrate this, let's say we have variable $$x$$ that is distributed as a standard Gaussian:

$$x \sim N(0,1)$$

The mean and variance are given by:

$$\mu = \mathbb{E}[x] = \int x p(x) dx = 0$$

$$\sigma^2 = \text{var}[x] = \mathbb{E}[(x - \mu)^2] = \mathbb{E}[x^2] - \mathbb{E}[x]^2 = 1$$

When we add $$c$$ to $$x$$, the mean shifts by $$c$$:

$$\mathbb{E}[x + c] = \mathbb{E}[x] + \mathbb{E}[c] = \mathbb{E}[x] + c$$

When we multiply $$x$$ by $$c$$, the variance multiplies with $$c$$:

$$\text{var}[cx] = \mathbb{E}[(cx)^2] -\mathbb{E}[cx]^2 = \mathbb{E}[c^2x^2] - (c\mathbb{E}[x])^2 = c^2(\mathbb{E}[x^2] - \mathbb{E}[x]^2) = c^2\text{var}[x]$$

These notes on shifting and scaling and on expectations might be helpfull to develop a deeper understanding.

• Thanks. I am an idiot. Feels like I forgot everything from my first statistics class.. However your links are very helpful. Commented Feb 10, 2019 at 16:04