Does anybody know how to find the confidence intervals for estimated parameters of a mixture of Gaussians by using EM algorithm?
This is a natural question but as such it has no answer: EM is an optimisation algorithm, not a statistical inference principle. As such, it returns a maximum likelihood point estimate of the parameter in the best case (or a local mode in the worse cases). To find confidence intervals on the parameters, you need to involve other statistical principles, like bootstrap or Bayesian inference. Note that in the case of mixtures standard approximations fail because of the lack of identifiability of the parameters and the degeneracy at the boundaries of the parameter space.
If this were me, I would formulate this as a Bayesian problem, like Xi'an said, so that confidence (well, credible) intervals fall out naturally. Since the mixture component means are random variables, the posterior distribution will tell you everything you need to know about your estimated parameters, beyond just point estimates and confidence intervals. While it's true that standard Bayesian methods like MCMC will perform poorly on mixture model data due to lack of identifiability and a highly multimodal posterior, you can mitigate this computationally (like by using Potentials in PyMC3) or by switching to Variational Inference, which is not only better suited to this problem but will also give you back a distribution for each parameter.