Substituting values from an association thread into a table of expressions

Question

I have asked a question here. Yet I am trying to do the same procedure with AssociationThread Suppose I have the following function which is a normal distribution:

σ = 4.75;
μ = 3.96;
f[x_] := 1/(σ*Sqrt[2 π]) Exp[-(1/2) ((x - μ)/σ)^2]

and I have the following list:

cases={2, 8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 28, 30, 34, 37, 43, \
55, 59, 90, 129, 180, 229, 276, 349, 396, 446, 571, 794, 1152, 1572, \
1965, 2282, 2679, 3225, 3905, 4814, 5745, 6826, 7860, 9068, 11084, \
13106, 15368, 17975, 20633, 23005, 25443, 28956, 32719, 36802, 40814, \
44865, 48188, 51221, 55551, 60024, 64256, 68271, 71776, 74771, 77456, \
80762, 84179, 87782, 91219, 94644, 97217, 99301, 102374, 105368, \
108129, 110897, 113666, 115476, 116880, 119100, 121384, 123731, \
125852, 127804, 129042, 130001, 131623, 133233, 134671, 136117, \
137141, 138034, 138759, 140004, 141102, 142177, 143070, 143979, \
144612, 145249, 146191, 147209, 148030, 148778, 149483, 149937, \
150322, 150832, 151375, 151917, 152371, 152806, 153147, 153452, \
153900, 154379, 154821, 155135, 155457, 155666, 155872, 156171, \
156481, 156846, 157120, 157391, 157652, 157860, 158139, 158447, \
158735, 158982, 159217, 159367, 159486, 159687, 159891, 160059, \
160209, 160325, 160382, 160391, 160391}
Prob = Table[NIntegrate[f[x], {x, i, i + 1}], {i, 1, 141}];
α = 
  AssociationThread[Table[Subscript[t, i], {i, 1, 141}] -> cases];
β = 
  AssociationThread[Table[Subscript[c, i], {i, 1, 141}] -> cases];
γ = 
  AssociationThread[Table[Subscript[p, i], {i, 1, 141}] -> Prob];

I want to replace the values from alpha, beta and gamma accordingly into the following table:

Table[1/(Subscript[c, j] - (1 - θ) Subscript[t, j])
   Sum[Subscript[p, 
    n] (Subscript[c, n] - (1 - θ) Subscript[t, n]), {n, 2, 
    141}], {j, 1, 140}]

I wonder how one can achieve this?

Answer 1

Take a smaller example with the same structure as the one in OP:

k = 4;
cc = Array[Subscript[c, #] &, k];
pp = Array[Subscript[p, #] &, k];
tt = Array[Subscript[t, #] &, k];

You can get your table using:

tbl = (1/Most[cc - (1 - θ) tt]) Rest[pp].Rest[cc - (1 - θ) tt];

TeXForm @ tbl

$\left\{\frac{p_2 \left(c_2-(1-\theta ) t_2\right)+p_3 \left(c_3-(1-\theta ) t_3\right)+p_4 \left(c_4-(1-\theta ) t_4\right)}{c_1-(1-\theta ) t_1},\\\frac{p_2 \left(c_2-(1-\theta ) t_2\right)+p_3 \left(c_3-(1-\theta ) t_3\right)+p_4 \left(c_4-(1-\theta ) t_4\right)}{c_2-(1-\theta ) t_2},\\\frac{p_2 \left(c_2-(1-\theta ) t_2\right)+p_3 \left(c_3-(1-\theta ) t_3\right)+p_4 \left(c_4-(1-\theta ) t_4\right)}{c_3-(1-\theta ) t_3}\right\}$

First, consider associations with symbolic values to see that replacements work as intended:

assoccc = AssociationThread[cc, Array[Subscript[Γ, #] &, k]];
assocpp = AssociationThread[pp, Array[Subscript[ρ, #] &, k]];
assoctt = AssociationThread[tt, Array[Subscript[τ, #] &, k]];

Join the three associations into a single association and use ReplaceAll:

assocjoin = Join[assoccc, assocpp, assoctt];

tbl /. assocjoin

$\left\{\frac{\rho _2 \left(\Gamma _2-(1-\theta ) \tau _2\right)+\rho _3 \left(\Gamma _3-(1-\theta ) \tau _3\right)+\rho _4 \left(\Gamma _4-(1-\theta ) \tau _4\right)}{\Gamma _1-(1-\theta ) \tau _1},\\\frac{\rho _2 \left(\Gamma _2-(1-\theta ) \tau _2\right)+\rho _3 \left(\Gamma _3-(1-\theta ) \tau _3\right)+\rho _4 \left(\Gamma _4-(1-\theta ) \tau _4\right)}{\Gamma _2-(1-\theta ) \tau _2},\\\frac{\rho _2 \left(\Gamma _2-(1-\theta ) \tau _2\right)+\rho _3 \left(\Gamma _3-(1-\theta ) \tau _3\right)+\rho _4 \left(\Gamma _4-(1-\theta ) \tau _4\right)}{\Gamma _3-(1-\theta ) \tau _3}\right\}$

For your example, use k = 141 and replace Array[Subscript[Γ, #] &, k] and Array[Subscript[τ, #] &, k] with cases and replace Array[Subscript[ρ, #] &, k] with Prob:

k = Length @ cases;
cc = Array[Subscript[c, #] &, k];
pp = Array[Subscript[p, #] &, k];
tt = Array[Subscript[t, #] &, k];
tbl = (1/Most[cc - (1 - θ) tt]) Rest[pp].Rest[cc - (1 - θ) tt];

α = AssociationThread[tt, cases];
β = AssociationThread[cc, cases];
γ = AssociationThread[pp, Prob];

tbl2 = tbl /. Join[α, β, γ];
Short[tbl2, 3]

Answer 2

From your code, it seems that $c=t=$cases, $p=$Prob. If so, $\theta$ is useless, since

$$ \frac{c_i-(1-\theta)t_i}{c_j-(1-\theta)t_j}=\frac{c_i-(1-\theta)c_i}{c_j-(1-\theta)c_j}=\frac{\theta c_i}{\theta c_j}=\frac{c_i}{c_j} $$

Then the code for your table is simply

n = Length[cases]; (* 141 *)

Rest[Inner[Times, Prob, cases, List]].ConstantArray[1/Most[cases], 
  n - 1]

or just

Rest[Prob].Rest[cases]/Most[cases]

If $c\neq t$, then similarly,

Rest[p].Rest[c - (1 - \[Theta]) t]/Most[c - (1 - \[Theta]) t]

Remember you don't need to create an Association in this case, just keep them in the vector (List).