Plot the probability density function

$\qquad w(x)=(1/(1+x^2))(1/x^{0.5})$ for $(0 < x < 1)$

Make up the subroutine flowchart to get the random numbers $x$, using the modified rejection technique with first factor used as the comparison function.

Use the flowchart to get a few random numbers in Wolfram Mathematica

Trial answer:

a = 0; b = 1;
f[x_] := (1/(1 + x^2)) (1/x^0.5)
h = 1; Nt = 1000000;
gr1 = Plot[f[x], {x, a, b}]

S := (
  x = y = b h; 
  While[y > f[x], 
    x = a + (b - a) RandomReal[]; 
    y = h RandomReal[]]; 
  Return[x];
)

rez = Table[S, {i, Nt}];
gr2 = Histogram[rez, Automatic, "PDF"]
Show[gr1, gr2]