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Integration of product of two quantile functions

Integrate[
 Quantile[NormalDistribution[0, 1], p] * 
  Quantile[StudentTDistribution[0, 1, 5], p] 
 , {p, 0, 1} ]

gives the a conditional expression in integrand.

Question Why does Mathematica not give a numerical result in this case?

Also, the domain is already set as $[0,1]$. Why does the following condition appear?

if 0<= p <= 1

The following code shows the expression I got when put in Mathematica.

Integrate[ConditionalExpression[(-Sqrt[2])*InverseErfc[2*p]*
       Piecewise[{{(-Sqrt[5])*
       Sqrt[-1 + 1/InverseBetaRegularized[2*p, 5/2, 1/2]], 
      0 < p < 1/2}, {0, p == 1/2}, 
           {Sqrt[5]*
       Sqrt[-1 + 1/InverseBetaRegularized[2*(1 - p), 5/2, 1/2]], 
      1/2 < p < 1}, {-Infinity, p <= 0}}, Infinity], 0 <= p <= 1], 
   {p, 0, 1}]
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  • $\begingroup$ Welcome to the Mathematica Stack Exchange. Try using NIntegrate for numerical integration. $\endgroup$
    – Syed
    Commented Nov 12, 2021 at 5:12

1 Answer 1

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integrand = 
 Simplify[
  Quantile[NormalDistribution[0, 1], p]*
    Quantile[StudentTDistribution[0, 1, 5], p] // PiecewiseExpand, 
  0 < p < 1]

enter image description here

NIntegrate[integrand, {p, 0, 1}]

(* 1.2695 *)
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