I encounter a problem which I thought I can handle, however, I struggle a lot with finding a solution:

The following setting applies: I want to compute the posterior probability of an event, which is defined as the fraction of two distinct arrival rates of, lets say, some agents. So I define as priors for the arrival rates $\mu$ and $\varepsilon$ Gamma distributions with parameters $\mu\sim G(\alpha_\mu,\beta)$ and $\varepsilon\sim G(\alpha_\varepsilon,\beta)$.

The event I am interested in depends on some constant $1>c>0$ and is characterized by $$P = \frac{c(\varepsilon+\mu)}{\varepsilon + c\mu}.$$ Well, the nominator takes the form $c(\varepsilon+\mu) \sim G(\alpha_\mu + \alpha_\varepsilon, \frac{\beta}{c})$, however, as the two parts in the denominator do not have the identical scale parameter I cannot simply identify the distribution of $P$ as a beta distribution as would be the case given I would simply compute $\frac{x}{y+x}$ with $x\sim G(\alpha_1,\theta), y\sim G(\alpha_2,\theta)$.

Is there any way to overcome this problem and to find the probability distribution of $P$? Or, is there another distribution which would give me some closed form solutions for $P$?

  • 3
    $\begingroup$ Algebra shows that $$P = 1 - \frac{1-c}{1 + c\frac{\mu}{\epsilon}}.$$ The ratio $\mu/\epsilon$ is a multiple of an $F$ distribution. That's probably as far as you'll get in simplifying things, but it gives you a direct, simple way to compute the CDF and inverse CDF of $P$. (Notice that $\beta$ plays no role; it drops out of the ratio that defines $P$.) $\endgroup$
    – whuber
    Jul 14, 2016 at 21:21
  • $\begingroup$ Great, thank you @whuber ! Do you have any idea whether I can say something about the moments of this representation? $\endgroup$
    – muffin1974
    Jul 15, 2016 at 8:41
  • 1
    $\begingroup$ Use the CDF directly. $\endgroup$
    – whuber
    Jul 15, 2016 at 13:20
  • $\begingroup$ @whuber What is two gamma variate are not independent with weak correlation (say 0.3)? Is the ratio still follow F? $\endgroup$ Oct 4, 2018 at 14:56
  • $\begingroup$ @Over probably not. You can see this by considering an extreme case where the correlation is nearly $1:$ that implies the ratio will be closely concentrated around $1$ itself, far more so than any $F$ distribution whose degrees of freedom are anywhere close to the DFs of the Gamma distributions. $\endgroup$
    – whuber
    Oct 4, 2018 at 17:05

1 Answer 1


You could think of approximating it using Kernel Density Estimation. Since you can easily simulate gamma variates, then you can easily simulate from P. Using a simulated sample, you can then construct a nonparametric estimator of this density. The following R code shows the approximation using 100,000 simulations (quite a few).

# Parameters
a.mu = 5
a.eps = 5
beta = 5
c = 0.5
# Number of simulations
N = 100000

# Simulated P
mu = rgamma(N,a.mu,beta)
eps = rgamma(N,a.eps,beta)
P.sim = c*(eps+mu)/(eps+c*mu) 

# Bandwidth
h = bw.nrd0(P.sim)

# Kernel density estimator
denP = Vectorize(function(t)  mean(dnorm(t-P.sim,0,h)) )

# Fit

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.