Continous Observation with HMM & Gaussian

Question

I've read that for HMMs with a discrete output we assign an emission probability $b_j(o_t)$ of emitting symbol $o_t$ when in state $S_j$. What I don't understand is why, in the case of continous outputs, we can set $b_j(o_t)$ to be a mixture of probability density functions (i.e. Gaussians). To my knowledge a PDF does not directly represent probabilites.