# Integration of product of two quantile functions gives a conditional expression in the integrand

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)*InverseErfc[2*p]*
Piecewise[{{(-Sqrt)*
Sqrt[-1 + 1/InverseBetaRegularized[2*p, 5/2, 1/2]],
0 < p < 1/2}, {0, p == 1/2},
{Sqrt*
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}]

• Welcome to the Mathematica Stack Exchange. Try using NIntegrate for numerical integration.
– Syed
Nov 12, 2021 at 5:12

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

(* 1.2695 *)