# Find a probability with random variates from the same distribution

I'm trying to find Probability involving 3 random variates from the same distribution, but Mathematica does not seem to solve it.

Probability[ z1 > 1.325*((z2)^2 + (z3)^2)^(1/2), {z1 \[Distributed]
NormalDistribution[0, 1],   z2 \[Distributed] NormalDistribution[0, 1],
z3 \[Distributed] NormalDistribution[0, 1]}]


What am I missing?

I also tried:

Probability[ z1 > 1.325*((z2)^2 + (z3)^2)^(1/2), {z1, z2, z3}
\[Distributed] NormalDistribution[0, 1]]

• NProbability[ z1 > 1325/100*Sqrt@((z2)^2 + (z3)^2), {z1 \[Distributed] NormalDistribution[0, 1], z2 \[Distributed] NormalDistribution[0, 1], z3 \[Distributed] NormalDistribution[0, 1]}] Commented Nov 30, 2014 at 20:08

As this is simple enough, you could use definitions :

dist = MultinormalDistribution[{0, 0, 0}, IdentityMatrix[3]] ;
prob[a_] = Integrate[PDF[dist, {z1, z2, z3}],
{z2, -Infinity, Infinity},
{z3, -Infinity, Infinity},
{z1, a Sqrt[z2^2 + z3^2], Infinity}]

(* ConditionalExpression[1/4 (2 - 2/Sqrt[1 + 1/a^2]), Re[a] > 0] *)


Then

prob[1.325]
(* 0.100905 *)