Suppose I have 300 daily returns that come from one of two distributions. The probability it comes from distribution 1 is 50% and the probability it comes from distribution 2 is 50% (i.e. equally weighted).
The first distribution is: X~N(0,0.01^2) The second distribution is: Y~N(0,0.02^2)
What is the resulting distribution?
If both distributions are normal I believe it should be Z = 0.5X+0.5Y -> Z~N(0,0.000125)
Is this correct? Any help is appreciated.