Consider two random complex numbers
z = u₁ + iv₁ and
w = u₂ + iv₂,
where u₁, v₁, u₂, v₂ are independent standard normal random variables (N(0,1)).
Then what is the probability distribution of the absolute value of the product |zw|?
Some empirical investigation (simulation) shows that the distribution looks like this:
Turns out an ancient paper(*) has the answer.
If z = u₁ + iv₁ and w = u₂ + iv₂, where u₁, u₂, v₁, v₂ ~ N(0,1) (and independent), then the probability density of
r := |wz|
is given by
where K₀ denotes the modified Bessel function of the second kind with order 0.
(*) Wells, Anderson, Cell (1962) "The Distribution of the Product of Two Central or Non-Central Chi-Square Variates"
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