I am studying this piecewise function:
Piecewise[{{f1[x, h], x < a}, {f2[x, a, h], a <= x < b},
{f3[x, a, b, h], b <= x < c}, {f4[x, a, b, c, h], x >= c}}]
where
f1[x_,h_] := pdf1[x]/pdf[x] (h + (1 - h) 2 cdf1[x])/(h + (1 - h) 2 cdf[x])
f2[x_,a_,h_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[a]))/
(h + (1 - h) 2 (1 - cdf[x] + cdf[a]))
f3[x_,a_,b_,h_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (cdf1[x] - cdf1[b] + cdf1[a]))/
(h + (1 - h) 2 (cdf[x] - cdf[b] + cdf[a]))
f4[x_,a_,b_,c_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[c] - cdf1[b] +
cdf1[a]))/(h + (1 - h) 2 (1 - cdf[x] + cdf[c] - cdf[b] + cdf[a]))
where, in turn
pdf1[x_] := PDF[NormalDistribution[1, 1], x];
pdf[x_] := PDF[NormalDistribution[0, 1], x];
cdf1[x_] := CDF[NormalDistribution[1, 1], x];
cdf[x_] := CDF[NormalDistribution[0, 1], x];
I want the first and the third piece of the piecewise function to be smaller than a certain value $l$, say $l=0.48$; and the second and fourth piece of the function to be greater than the same $l$. The following code represents the intervals of $a$, $b$ and $c$ such that the four inequalities are satisfied:
{amin = NArgMin[{f2[a], f2[a] > l}, a], amax = NArgMax[{f1[a], f1[a] < l}, a],
a = RandomReal[{amin, amax}]}
{bmin = a, bmax = NArgMax[{f3[b], f3[b] < l}, b], b = RandomReal[{bmin, bmax}]}
{cmin = b, cmax = NArgMax[{f3[c], f3[c] < l}, c], c = RandomReal[{cmin, cmax}]}
Note however that, for example, you have to give a random value to $a$ since the lower bound of the interval of the $b$'s depends on $a$.
I would like to plot in a 3D Graph all the possible combinations $(a,b,c)$ such that the system of inequalities is satisfied. Moreover, I would like to implement the Manipulate command so that I can study how the set of all possible combinations $(a,b,c)$ changes as $h$ varies between 0 and 1. If a 3D graph was not feasible, actually any graphical representation of the set of solutions of $(a,b,c)$ would be fine anyway. Thank you if you can help.
RegionPlot3D
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