#!/usr/bin/env wolframscript (* ::Package:: *) \!\( \(\*SubsuperscriptBox[\(f\), SubscriptBox[\(X\), \(k\)], \(-\)]\)[x]\) = \!\( \*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]\( \*FractionBox[\(1\), \( \*SubscriptBox[\(\[Sigma]\), SubscriptBox[\(Q\), \(k\)]] \*SqrtBox[\(2 \[Pi]\)]\)] \*SuperscriptBox[\(E\), \(- \*FractionBox[ SuperscriptBox[\((x - F*v - B* \*SubscriptBox[\(u\), \(k\)])\), \(2\)], \(2 \*SuperscriptBox[ SubscriptBox[\(\[Sigma]\), SubscriptBox[\(Q\), \(k\)]], \(2\)]\)]\)]* \*FractionBox[\(1\), \( \*SubsuperscriptBox[\(\[Sigma]\), \(k - one\), \(+\)] \*SqrtBox[\(2 \[Pi]\)]\)] \*SuperscriptBox[\(E\), \(- \*FractionBox[ SuperscriptBox[\((v - \*SubsuperscriptBox[\(\[Micro]\), \(k - one\), \(+\)])\), \(2\)], \(2 \*SuperscriptBox[ SubsuperscriptBox[\(\[Sigma]\), \(k - one\), \(+\)], \(2\)]\)]\)] \[DifferentialD]v\)\) \!\( \(\*SubsuperscriptBox[\(f\), SubscriptBox[\(X\), \(k\)], \(+\)]\)[x]\) = Simplify[(\!\( \*SubsuperscriptBox[\(\[Integral]\), \(-\[Infinity]\), \(\[Infinity]\)]\( \*FractionBox[\(1\), \( \*SubscriptBox[\(\[Sigma]\), SubscriptBox[\(R\), \(k\)]] \*SqrtBox[\(2 \[Pi]\)]\)] \*SuperscriptBox[\(E\), \(- \*FractionBox[ SuperscriptBox[\(( \*SubscriptBox[\(y\), \(k\)] - H*x)\), \(2\)], \(2 \*SuperscriptBox[ SubscriptBox[\(\[Sigma]\), SubscriptBox[\(R\), \(k\)]], \(2\)]\)]\)]* \*FractionBox[\(1\), \( \*SubsuperscriptBox[\(\[Sigma]\), \(k\), \(-\)] \*SqrtBox[\(2 \[Pi]\)]\)] \*SuperscriptBox[\(E\), \(- \*FractionBox[ SuperscriptBox[\((x - \*SubsuperscriptBox[\(\[Micro]\), \(k\), \(-\)])\), \(2\)], \(2 \*SuperscriptBox[ SubsuperscriptBox[\(\[Sigma]\), \(k\), \(-\)], \(2\)]\)]\)] \[DifferentialD]x\)\))^-1*1/(Subscript[\[Sigma], Subscript[R, k]] Sqrt[2\[Pi]]) E^(-((Subscript[y, k]-H*x)^2/(2Subscript[\[Sigma], Subscript[R, k]]^2)))*1/(\!\( \*SubsuperscriptBox[\(\[Sigma]\), \(k\), \(-\)] \*SqrtBox[\(2 \[Pi]\)]\)) E^(-((x-\!\(\*SubsuperscriptBox[\(\[Micro]\), \(k\), \(-\)]\))^2/(2\!\(\*SubsuperscriptBox[\(\[Sigma]\), \(k\), \(-\)]\)^2)))]