Get distibution depending on three random variables

Question

Given the function $\text{result} = \frac{a*b}{1-c}$ with 3 independent equally distributed random variables $a$, $b$, and $c$, how do I derive the distribution of $\text{result}$? How can I get the range of the top 20 percent of $\text{result}$? I did it all in Excel to get approximations, but my math is a bit rusted.

Answer 1

You say you have three independent, "equally distributed" random variables, which I will call $X_i,$ for $i = 1,2,3.$. I take this to mean that the three random variables have the same distribution. Whether your question makes practical sense depends on what distribution that is.

You want to know about the distribution of $$ W = \frac{X_1X_2}{1-X_3}.$$ Notice that if the distribution of $X_3$ puts probability in a neighborhood of $1,$ then $W$ will not have a mean. But you are interested in the 80th percentile of $W,$ so let's use simulation in R statistical software to look at four distributions for the $X_i.$

(1) $X_i \stackrel{iid}{\sim} Unif(0,1).$ In this case, the 80th quantile of $W$ is about 1.25. Its distribution has a long heavy tail to the right owing to values of $X_3$ near 1. Accordingly, the histogram in panel (1) of the figure below is truncated to omit values above 1.2.

m = 10^6
x1 = runif(m);  x2 = runif(m);  x3 = runif(m)
w = x1 * x2/(1 - x3)
quantile(w, .8)
##      80% 
## 1.249817 

(2) $X_i \stackrel{iid}{\sim} Unif(2,3).$ In this case, the 80th quantile of $W$ is about -3.33, and $E(W) \approx -4.33.$ The histogram in panel (2) shows a moderately left-skewed distribution, which one ought to be able to derive analytically.

(3) $X_i \stackrel{iid}{\sim} Norm(0,1).$ In this case, the 80th quantile of $W$ is about 0.58. Its distribution has long heavy tails in both directions, owing to values of $X_3$ near 1. Accordingly, the histogram in panel (1) of the figure below is truncated to show only values with $|W| < 0.6.$

(4) $X_i$ have a 'delayed exponential distribution'. That is, $X_i = Y_i + 3,$ where $Y_i \sim Exp(rate=1).$ In this case, the 80th quantile of $W$ is about -3.79, and $E(W) \approx -5.78.$ The denominator of $W$ takes negative values separated from 0. The histogram in panel (4) shows left-skewed distribution with a thin left tail, and one ought to be able to derive the distribution analytically.