p[e_]=Integrate[BesselK[5/3,x],{x,e,Infinity},Assumptions->e>0]

Plot[p[x]Sqrt[x] Exp[x],{x,5,15}]

alpha=1/N[Integrate[p[x],{x,0,100}],16]

pp[e_]=Integrate[alpha p[x],{x,0,e},Assumptions->e>2]

Plot[(1-pp[x])Sqrt[x]Exp[x],{x,5,15}]

Plot[(1-pp[x]) Sqrt[x] Exp[x]/(1+1/(9x)),{x,5,15}]

q[r_]=InverseFunction[pp][r]

q[0.9999999]

q'[0.999]

Plot[q[x],{x,0.990,0.999},BaseStyle->{PrintPrecision->6}]

NumberForm[Map[q,Range[0.0,0.8,0.02]],8]

NumberForm[Map[q',Range[0.0,0.8,0.02]],8]

NumberForm[Map[q,Range[0.80,0.99,0.01]],8]

NumberForm[Map[q',Range[0.80,0.99,0.01]],8]

NumberForm[Map[q,Range[0.990,0.999,0.001]],8]

NumberForm[Map[q',Range[0.990,0.999,0.001]],8]

NumberForm[Map[q,Range[0.9990,0.9999,0.0001]],8]

NumberForm[Map[q',Range[0.9990,0.9999,0.0001]],8]

NumberForm[Map[q,Range[0.99990,0.99999,0.00001]],8]

NumberForm[Map[q',Range[0.99990,0.99999,0.00001]],8]

NumberForm[Map[q,Range[0.999990,0.999999,0.000001]],8]

NumberForm[Map[q',Range[0.999990,0.999999,0.000001]],8]

NumberForm[Map[q,Range[0.9999990,0.9999999,0.0000001]],8]

NumberForm[Map[q',Range[0.9999990,0.9999999,0.0000001]],8]

al=0.000001Sqrt[13.403243]Exp[13.403243]

f[x_]=al Exp[-x]/Sqrt[x]

((2.4241496` (E^(-x)))/(Sqrt[x]))

f[31.4]