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.

Answer 1

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.

Answer 2

$N(x|\mu, \sigma^2)$ 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 $\mu, \sigma^2$.