Distribution of a function of exponential and uniform random variables? [closed]

Consider the following independent random variables:

\begin{equation} \begin{aligned} A &\sim \text{Exp}(\lambda_1), \\[6pt] C &\sim \text{Exp}(\lambda_2), \\[6pt] B &\sim \text{U}(0,1), \\[6pt] D &\sim \text{U}(0,1). \\[6pt] \end{aligned} \end{equation}

Define the random variable:

$$M \equiv \frac{(A+BC)^2}{AC(B-1)^2 + CD + A + C}.$$

How do I find the distribution of this random variable?

closed as off-topic by gung♦Dec 23 '18 at 19:18

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Self-study questions (including textbook exercises, old exam papers, and homework) that seek to understand the concepts are welcome, but those that demand a solution need to indicate clearly at what step help or advice are needed. For help writing a good self-study question, please visit the meta pages." – gung
If this question can be reworded to fit the rules in the help center, please edit the question.

• Could you explain where $M$ comes from? It is so strange (a non-homogeneous rational function of degree four) that one wonders whether it might be an attempt to solve a problem in an unusual way and that (therefore) a better way might be available. It is highly unlikely you will obtain any simple closed form for the distribution of $M.$ – whuber Dec 17 '18 at 18:48
• Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. – gung Dec 23 '18 at 19:17

Simulate it (this is python)

import numpy as np
lambda1 = 1.0
N = 1000000
A = np.random.exponential(scale=lambda1,size=N)
B = np.random.uniform(size=N)
C = np.random.exponential(scale=lambda1,size=N)
D = np.random.uniform(size=N)
M = (A+B*C)**2/(A*C*(B-1)**2 + C*D + A + C) # N realizations of the random variable

import matplotlib.pyplot as plt
_ = plt.hist(M, bins=5000, density=True) # plot

Here it is. You could regress something to it for practical purposes. The other way would be to use your rules for combinations of random variables and work really hard. I don't know what that would become • I need a mathematical expression for the pdf(M). – user231400 Dec 17 '18 at 16:11