how to estimate parameter in finite mixture of two different distribution using r code?

I have a data set for two distributions and I want to program the prediction function for both distributions

Your question is too general. You need to provide an example of your data and the equations for the distributions you want to fit.

Maybe Google something like "R fit distribution to data" which leads me to this package:

https://cran.r-project.org/web/packages/fitdistrplus/index.html

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