EM Algorithm - E step In the E-step, why are there 2 different thetas ? Isnt the expectation of a function $ E[x] = \int xp(x) ?$ If I know x then I would know p(x) as well right ? Based on what I am reading $ E[x] = \int xp(x') $. Can someone explain to me what is going on ? 
Also, how do I maximize the expectation of a gaussian function ? Solving the integral gives me the solution, i.e. the mean of the gaussian. I have no variable left like what is doing in the maximization step in the EM algorithm.

 A: First of all you have a function $Q(\Theta, \Theta^{(t)})$ that depends on two different thetas: $\Theta$ which is the new one. One strategy could be to insert multiple $\Theta$ into the function and take the one with the maximal value for example. Then there is $\Theta^{(t)}$. This is the 'current' estimate of the parameters. The idea of the EM Algorithm is to proceed from the current estimate of the parameters $\Theta^{(t)}$ to (basically, any) new estimation $\Theta$ that 'improves the situation'. Now the situation means to make the likelihood
$$p(x|\cdot) = \sum_z p(x,z|\cdot)$$
better than before, i.e. the goal is to make a step from $\Theta^{(t)}$ to $\Theta$ such that
$$p(x|\Theta) > p(x|\Theta^{(t)})$$
if notation confuses you then you can safely redefine $\Theta^{(t)}$ to $\Theta^{(\text{old})}$ and $\Theta$ to $\Theta^{(\text{new})}$.
Remark: There is a concept of 'conditional expectation' which is something like a generalization of the usual expected value you are referring to but it is NOT THE SAME! The object the EM algorithm is referring to is
  $$E[\log p(x,Z|\Theta) | X=x]$$
this is a really complicated object! It is the factorization of the random variable $E[\log p(x,Z|\Theta) | \sigma(X)]$ at $x$. 
However, one can show mathematically that
  $$E[g(x,Z)|X=x] = \int_{\mathcal{Z}} g(x,z) p(x|z) dy$$
Also, the usual notation might be confusing. Let us write $p_\Theta(...) = p(...|\Theta)$. Then what we want is
  $$Q(\Theta|\Theta^{(\text{old})}) = E[\log p_{\Theta}(x,Z^{\Theta^{(\text{old})}}) | X^{\Theta^{(\text{old})}}=x]$$
i.e. the density is the one of the random variables with the 'new' density but the actual random variables that we put in are the ones with the 'old' parameter. Hence,
$$E[\log p_{\Theta}(x,Z^{\Theta^{(\text{old})}}) | X^{\Theta^{(\text{old})}}=x] = \int_{\mathcal{Z}} p_\Theta(x,z) p_{\Theta^{(\text{old})}}(x|z) dz = \sum_z p_\Theta(x,z) p_{\Theta^{(\text{old})}}(x|z)$$
For the Gaussian mixture model you should study https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm#Gaussian_mixture.
