To determine distribution function of a quantity using other distributions

Question

Let $A=B/C$ where distributions of $B$ and $C$ are known to follow exponential distributions. Is there a method to determine distribution function of $A$ using the knowledge of the distributions of $B$ and $C$?

Answer 1

You are looking for a Ratio Distribution between two exponential distributions.   This requires knowledge of the joint distribution of the variables (ie: how-or-if the random variables are related).

If $A=B/C$ then $f_A(x) = {\displaystyle\int_\Bbb R} \lvert{z}\rvert\, f_{B,C}(zx, z)\operatorname d z$

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In the special case where $B,C$ are independent exponential r.v. with rate parameters $\lambda_B, \lambda_C$, then $$f_A(x) ~{= \mathbf 1_{x\in(0;\infty)}\cdot{\displaystyle\int_0^\infty} \lambda_B\lambda_C z\, \mathsf e^{-(\lambda_B x+\lambda_C)z}\operatorname d z \\ = \dfrac{\lambda_B\lambda_C}{(\lambda_B x+\lambda_C)^2}~\mathbf 1_{x\in(0;\infty)} }$$