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I am going to perform an action, $A$, on a system. The state of the system after the action can be $a_1$, with probability $p_1$, or $a_2$, with probability $p_2$. If the state of the system is $a_1$ ($a_2$), the reliability of the system is $x_1$ ($x_2$), both of which are normally distributed. I want to know the reliability (total mean and variance) of the system after doing the action $A$.

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Amy
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  • It's not clear to me what you're asking. What do you mean by "the probability of occurrence" and the "total mean and variance"? – jld May 23 '17 at 19:33
  • I am going to do an action which its output can be X1 with probability p1 or X2 with probability p2. the X1 and X2 are two normal continuous variables that have their own mean and variance. I want to know the mean and variance of my action. – Amy May 23 '17 at 21:24
  • Please edit your comment into your question, though I believe you mean "the mean and variance of the reliability that results from my action". This will be the mean and variance of a mixture distribution with mixing probabilities $p_i$. – Glen_b May 25 '17 at 00:39
  • whuber's answer at the indicated duplicate is general -- it doesn't rely on normality. – Glen_b May 25 '17 at 00:44

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