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Let $X \sim Rician(\mu,1)$. I want to ask about the distribution of product of $X$ as

  • $Y = aX$.
  • $Z = b - aX$.

From my intuition, $Y$ might also be the Rician distribution with parameter as $(a \mu, a^{2})$. However, I am not really sure about this one since I can not derive the characteristic function of $Y$.

As for $Z$, I completely have no idea how to begin.

I just begin studying about Rician distribution so I hope someone can give me some ideas or hints about this problem. Thank you very much.

M.bara
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1 Answers1

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Hint:

Start with computing $\mathbb{P}(Y \leq t)=\mathbb{P}(X \leq \frac t a)=\int_0^{\frac t a}f(x,\mu,1)dx=\frac{1}{a}\int_0^tf(\frac{y}{a},\mu,1)dy$ when $a>0$ and study $\frac{1}{a}f(\frac{y}{a},\mu,1)$.

The same reasoning should work also for $Z$

Filippo Giovagnini
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