I got a very comprehensive solution from  @janhardo and @sand15, who gave the most complete version.

Indeed, my understanding of the problem was wrong and presented solution (and discussion) teaches me a lot.

Thank you again.

@Carl Love Oh, yes, wX has variance w^2*a^2 , my mistake.

However the question remains how to define a mixture of two Gaussians with zero mean and variances a, and b respectively, with weight w, for which  

Var(Z)=w*a^2+(1-w)*b^2 

equivalent to Mathematica's MixtureDistribution function.

@Preben Alsholm 

Regarding my point 1.

Integral should evaluate to Sum((p__m)^2, m=0..inifinity) because

that's what I meant.

Regarding my point 2, it is still open question, I want to just change dead index I am summing on.

Thanks.

@nm,

Your experience is not common. Depending on your license you may have option to install second instance of maple on another computer. using the same purchasing code. Maybe it is good idea to not to install/disable antyvirus temporarily during trials.

Hope it helps!

Nothing changed, The problem still occurs in Win 10 Home 64 bit.

@tomleslie 

Forgot to add.

Path to "Physics Updates" has changed in 2020. it is now .\maple\toolbox\Physics Updates\lib

In 2019 it was in .\maple\toolbox\2019\Physics Updates\lib

@tomleslie 

It doesn't matter whether you have 64 or 32, Win 7 or Win 10. I suspect OS updates come with security descriptors which prevent maple from getting permission to your folder. No hope, copy "by hand" and it will work. 

Congratulations. My 7 years old doughter plays with it third day. Thank you.

BR,

JW

In fact, after 2019.1 update I can update Physics in GUI. 

 I have exactly the same symptoms. Win 10 Home.