What does $N(x|\mu, \sigma^2)$ mean?

Question

I am supposed to show that $f(x) = \sum_{k=1}^{K}\pi_k N(x|\mu_k, \sigma_{k}^2)$ complies with the properties of a density function but I have no idea how to do this since I am not sure what $N(x|\mu_k, \sigma_{k}^2)$ means.

I know $X \sim N(\mu, \sigma^2)$ means that the random variable X follows a normal distribution with mean $\mu$ and variance $\sigma^2$. I'm just not sure how $x|\mu_k$ changes things.

This is probably a very silly question but your help will be appreciated.

It's worth nothing that this is not standard notation, so feel vindicated in your confusion. — Cliff AB, Feb 19 '20 at 03:30
@CliffAB what is the standard notation for this? I came to this asking the same question as the OP — Matthew Barlowe, May 09 '21 at 19:17

score 6 · Answer 1 · edited Jun 01 '20 at 16:01

6

$N(x|\mu, \sigma^2)$ combines the two notations: $x \sim N(\mu, \sigma^2)$ and $p(x| \mu, \sigma^2)$. So it reads: $x$ is normally distributed with parameters $\mu, \sigma^2$.

edited Jun 01 '20 at 16:01

MJimitater

269

answered Feb 18 '20 at 20:22

Tim

138,066

score 6 · Accepted Answer · answered Feb 18 '20 at 20:23

6

Here, it means the normal PDF: $$\mathcal{N}(x|\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/2\sigma^2}$$

The $\mu,\sigma^2$ in given side means that you can treat them as known quantities.

answered Feb 18 '20 at 20:23

gunes

57,205

Ooh ok I see. Thought there was something more to it. Thank you! – Susan-l3p Feb 18 '20 at 20:28

What does $N(x|\mu, \sigma^2)$ mean?

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