# What are the meanings of $\langle\cdot,\cdot\rangle$ operator

I used to know that $\langle a,b\rangle$ is an inner product of two vectors $a$ and $b$. But recently in a research paper I found that during solution of birth death model of a Markov random process, it is written that, The rate equation of markov random process is

$$\frac {dP_N}{dN}=\mu(N+1)P_N+1 -[(\lambda + \mu)N+v]P_N +[\lambda(N-1)+v]P_N-1) \$$

partial differential equation for the moment generating function

$$Q(z,t)=\langle(1-z)^N\rangle=\sum\limits_{i=1}^\infty (1-z)^N. P_N(t)\$$ may be deduced from the first rate equation, $$\frac {\delta Q}{\delta t}=z[-\mu +\lambda(1-z)]\frac{Q}{z}-vzQ ,$$

i) Here, I can not understand what this symbols actually meaning? can <> symbol may be used to find statistical average also?? ii) How this derivation can be done??

• In addition to the answer, be aware that there are other meanings in other contexts (example) – Pedro May 2 '18 at 2:55

Notice that it only takes in one input $\langle \cdot \rangle$.