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

1 Answer 1

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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 *)
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