FindInstance on a system with three variables keeps evaluating

Question

I have been trying to find a set of parameters for a function that fit some constraints.

FindInstance[
 -a*PDF[NormalDistribution[24, σ], 1] + b == 20 &&
 -a*PDF[NormalDistribution[24, σ], 24] + b == 3 &&
 Total[Table[-a*PDF[NormalDistribution[24, σ], x] + b, {x, 47}]] == 350,
 {a, σ, b}, Reals]

I am trying to solve for a function in the form of

-a*PDF[NormalDistribution[24., σ], x] + b

with the following constraints:

-a*PDF[NormalDistribution[24., σ], 1] + b == 20
-a*PDF[NormalDistribution[24., σ], 24] + b == 3
Total[Table[-a*PDF[NormalDistribution[24, σ], x] + b, {x, 47}]] == 350

The cell has been evaluating for 26 hours and still hasn't given me a result. Could you please guide me on how to edit the code to give me a faster result?

Answer 1

$Version

(* "14.1.0 for Mac OS X ARM (64-bit) (July 16, 2024)" *)

Clear["Global`*"]

sol1 = NSolve[{
   -a*PDF[NormalDistribution[24, σ], 1] + b == 20,
   -a*PDF[NormalDistribution[24, σ], 24] + b == 3,
   Total[Table[-a*PDF[NormalDistribution[24, σ], x] + b, {x, 47}]] == 
    350, σ > 0}, {a, b, σ}, Reals, WorkingPrecision -> 50]

(* {} *)

There is no solution. Reformulate as a minimization.

obj = Total[(SubtractSides[{
        -a*PDF[NormalDistribution[24, σ], 1] + b == 20,
        -a*PDF[NormalDistribution[24, σ], 24] + b == 3,
        Total[
          Table[-a*PDF[NormalDistribution[24, σ], x] + b, {x, 47}]] == 
         350}][[All, 1]])^2];

The approximate solution is then

sol2 = N[NMinimize[{obj, σ > 0}, {a, b, σ}, 
   WorkingPrecision -> 50,
   Method -> "NelderMead"]]

(* {4.02864, {a -> 381531., b -> 1164.26, σ -> 130.873}} *)