10^Log[50, 1/(1 - Erf[5.33])] // N

Plot[Erf[x], {x, -2, 2}]

80. /15

Log[2, 20*(10^12)] // N

InverseErf[1 - 1/(100. *(10^8))]

100 - 4.8129*15

RSolve[

Quit

eqn := {y[n + 1] == 1.05*y[n]  + 0.05* 10*(1.05^n), 
        y[0] == 1}

Simplify[RSolve[eqn, y[n], n]]

\(Plot[
      1.001`\^n\ \((1 + 9.523809523809524`\ n)\), {n, 0, 1000}]\)

Table[

ListContourPlot[
        Table[(x^2) + (y^2) + Random[Real, {-0.75, 0.75}], {x, -2, 2, 
            0.1}, {y, -2, 2, 0.1}]];

Table[i, {i, 1, 10}]

Normal

<< Statistics`NormalDistribution`

ndist = NormalDistribution[100, 15]

pdf = PDF[ndist, x]

1/(pdf /. x -> 20) // 
      NU . S . foreign debt shows its teeth

Erf

cdf = CDF[ndist, x]

1/(cdf /. x -> 15) // N

Quantile[ndist, (10^(-11))] // N

Quit

RSolve[{d[i + 1] == g[i + 1] + d[i]*1.05, g[i + 1] == g[i]*1.05, 
        g[0] == 1, d[0] == 0}, {g[i], d[i]}, i]

Plot[((1.06^x) - (1.06^x))/(1.06^x), {x, 1, 100}]

Limit[(((((1 + x))^5) - 1)/(1 + x - 1)), x -> 0]

Plot[{Log[1 + Exp[x]], 
        Exp[x]/(1 + Exp[x])}, {x, -10, -2}]

Plot[{Log[1 + Exp[x]], Exp[(x^3)]}, {x, -10, -1}, 
      PlotRange -> {0, 0.14}]

ContourPlot[
      Log[(1 + Exp[x1]) (1 + Exp[x2])], {x1, -2, 2}, {x2, -2, 
        2}]

Plot[Exp[x]/(1 + Exp[x]), {x, -5, 5}]

Quit

eqn = {(((t1 + t1 t2 t3))/(1 + t1 + t2 + t1 t2 t3)) == 
          m1,  (((t2 + t1 t2 t3))/(1 + t1 + t2 + t1 t2 t3)) == 
          m2,  (((t1 t2 t3))/(1 + t1 + t2 + t1 t2 t3)) == 
          m3}

Solve[eqn, {t1, t2, t3}] // Simplify

t1  + t1 t2 t3  =  m1 Z,  t2 + t1 t2 t3 = m2 Z, 
    t1 t2 t3  =  m3 Z where Z = 
        1 + t1 + t2 + t1 t2 t3

eqn2 = {(((t1 +  t3))/(1 + t1 + t2 +  t3)) == 
          m1,  (((t2 +  t3))/(1 + t1 + t2 +  t3)) == 
          m2,  (( t3)/(1 + t1 + t2 +  t3)) == 
          m3}

Solve[eqn2, {t1, t2, t3}] // Simplify

eqn3 = {(t0/(1 + t0)) == 
          m0, (((t0 t3))/(1 + t0 t3)) == 
          m1, (((t0 t4))/(1 + t0 t4)) == 
          m2, (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6)) == 
          m3}

Solve[eqn3, {t0, t3, t4, t6}] // Simplify

3 Variable no 3-way logistic regression

eq1 = (t0/(1 + t0)) p00 + (((t0 t3))/(1 + t0 t3)) 
            p10 + (((t0 t4))/(1 + t0 t4)) 
            p01 + (((t0 t3 t4))/(1 + t0 t3 t4)) p11 == 
        m0

eq2  =  (((t0 t3))/(1 + t0 t3)) 
            p10 + (((t0 t3 t4))/(1 + t0 t3 t4)) p11 == 
        m3

eq3 = (((t0 t4))/(1 + t0 t4)) 
            p01 + (((t0 t3 t4))/(1 + t0 t3 t4)) p11 == 
        m4

Solve[{eq1, eq2, eq3}, {t0, t3, t4}]

{eq1, eq2, eq3}

Quit

Solve[{(t0/(1 + t0))*p00 + ((t0*t3)/(1 + t0*t3))*p10 + ((t0*t4)/(1 + t0*t4))*p01 + 
     ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == m0, 
   ((t0*t3)/(1 + t0*t3))*p10 + ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == m3, 
   ((t0*t4)/(1 + t0*t4))*p01 + ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == m4}, {t0, t3, t4}]

Solve[{(t0/(1 + t0))*p00 + ((t0*t3)/(1 + t0*t3))*
              p10 + ((t0*t4)/(1 + t0*t4))*
              p01 + ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == 
          m0, ((t0*t3)/(1 + t0*t3))*
              p10 + ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == 
          m3, ((t0*t4)/(1 + t0*t4))*
              p01 + ((t0*t3*t4)/(1 + t0*t3*t4))*p11 == 
          m4}, {t0, t3, t4}]

Simplified Version

eqns = {(( t0)/(1 + t0)) + (( t0 t3)/(1 + t0 t3)) + (( t0 t4)/(1 + t0 t4)) + (( t0 t3 t4)/(1 + t0 t3 t4)) == 
          m0, (( t0 t3)/(1 + t0 t3)) + (( t0 t3 t4)/(1 + t0 t3 t4)) == 
          m3, (( t0 t4)/(1 + t0 t4)) + (( t0 t3 t4)/(1 + t0 t3 t4)) == m4}

Solve[eqns, {t0, t3, t4}]

Log - likelihood for logistic regression model

4 variable Logistic Regression model

eqns = {(t0/(1 + t0)) + (((t0 t3))/(1 + t0 t3)) + (((t0 t4))/(1 + t0 t4)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6)) == 
          4 m0, (((t0 t3))/(1 + t0 t3)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6)) == 
          4 m3, (((t0 t4))/(1 + t0 t4)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6)) == 
          4 m4, (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6)) == 
          4 m6}

Solve[eqns, {t0, t3, t4, t6}]

Solution

sol = {t0 -> (((m0 - m3 - m4 + m6))/(1/4 - (m0 - m3 - m4 + m6))), t3 -> (((m3 - m6))/(1/
                    4 - (m3 - 
                      m6))) ((1/4 - (m0 - m3 - m4 + m6))/((m0 - m3 - m4 + m6))), 
        t4 -> (((m4 - m6))/(1/
                    4 - (m4 - 
                      m6))) ((1/4 - (m0 - m3 - m4 + m6))/((m0 - m3 - m4 + m6))), 
        t6 -> (m6/(1/4 - 
                  m6)) ((((m0 - m3 - m4 + m6))/(1/4 - (m0 - m3 - m4 + m6)))) ((1/4 - (m3 - m6))/((m3 - 
                    m6))) ((1/4 - (m4 - m6))/((m4 - m6)))}

eqns /. sol // Simplify

expr = {(1/4) ((t0/(1 + t0)) + (((t0 t3))/(1 + t0 t3)) + (((t0 t4))/(1 + t0 t4)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6))), (1/4) ((((t0 t3))/(1 + t0 t3)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6))), (1/4) ((((t0 t4))/(1 + t0 t4)) + (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6))), (1/4) (((t0 t3 t4 t6))/(1 + t0 t3 t4 t6))}

expr /. sol // Simplify

Plus[{d, 1}]

\(a1 = \(x\_# &\) /@ Table[i, {i, 1, 2}]\)

a2 = Subsets[a1]

a3 = Prepend[#, 1] & /@ a2

a4 = Times[#] & /@ a3

a4 = MapIndexed[List, a3]

Table[i, {i, 0, 3}]

Thread[{{1, (Subscript[x, 1]), (Subscript[x, 2]), (Subscript[x, 1]) (Subscript[x, 2])}, {0, 1, 2, 3}}]

Table[a8, Evaluate[Sequence @@ bounds]]

Table[a8, Sequence @@ bounds]

Flatten[a9]