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I know that the sum of two Wrapped Normal Variables is a Wrapped Normal Variable. In particular, if $\theta_1 \sim WN(\mu_1, \rho_1)$ and $\theta_2 \sim WN(\mu_2, \rho_2)$, then $\theta_1 + \theta_2 \sim N(\mu_1+\mu_2, \rho_1\rho_2)$ (Jammalamadaka & SenGupta, 2001; p. 44).

Is the product of two Wrapped Normal Variables a Wrapped Normal Variable? If so, what is the relationship of the initial means and variances with the final mean and variance?

honi
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    I have no idea what a Wrapped Normal Distribution is and I have no interest in finding out either, but as a general rule of thumb, properties that apply to sums hardly ever carry over to products, and I would be astonished to find out that the product of Wrapped Normal Random Variables is a Wrapped Normal Random Variable. For _independent_ random variables (of any kind, not just Wrapped Normals), $E[XY]=E[X]E[Y]$ while the variance is $${\rm Var}(XY)=E(X^2Y^2)− (E(XY))^2={\rm Var}(X){\rm Var}(Y)+{\rm Var}(X)(E(Y))^2+{\rm Var}(Y)(E(X))^2$$ cf. https://stats.stackexchange.com/q/52646/6633 – Dilip Sarwate Dec 22 '18 at 04:16
  • https://en.wikipedia.org/wiki/Wrapped_normal_distribution – kjetil b halvorsen Dec 24 '18 at 12:38
  • Could you please explain what form of "product" you mean? Because the $\theta_i$ are defined only modulo $2\pi,$ the usual multiplication of numbers is not even defined! – whuber Dec 24 '18 at 16:05
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    you are right. i failed to fully understand what i was asking in order to ask the right question. i will ask another question to get at what i am actually interested in knowing. – honi Dec 24 '18 at 16:07

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