Probability Distribution of q: Why and What is \langle \cdot \rangle_P?

[LagrangeEuler]

If $$P(q_1,q_2)$$ is probability distribution for two random variables $$q_1,q_2$$, let us define
$$q=\frac{q_1+q_2}{2}$$

Probability distribution for q is then

$$P'(q)=\int dq_1dq_2\delta (q-\frac{1}{2}(q_1+q_2)P(q_1,q_2)=\langle \delta (q-\frac{1}{2}(q_1+q_2) \rangle_P$$

Why?

What is label $$\langle \cdot \rangle_P$$ exactly?

 

[chiro]

Hey LagrangeEuler and welcome to the forums.

The symbol refers to the expectation with respect to the random variable P. This notation is used a lot in physics when denoted the expectation of a random variable.

You also get stuff involving the other bra-ket things like inner products and the use of operators and vector products like <a|b> and |a><b| which is known as Dirac notation.

 

[Bacle2]

More generally, there are general methods for calculating the distribution of a sum of random variables, depending the distribution of the individual variables:

http://www.cuil.pt/r.php?cx=0046045...stribution+of+the+sum+of+two+random+variables

Is that your question?