# Probability density of compound triangular distribution with uniformly distributed mode?

What are the probability density function and cumulative distribution function of a compound triangular distribution with uniformly distributed mode, both supported on $(-a, a)$? I.e.,

$$m \sim \mathrm{Unif}(-a, a) \\ X \sim \mathrm{Triang}(-a, a, m)$$

Does it have a name?

It is not a Wigner semicircle distribution and it is close to a scaled, shifted beta distribution with $\alpha = \beta \approx 1.75$, but I'm interested in whether it has a simple closed form.

• Yes, the PDF and CDF have closed forms. Indeed, for $a=1$ (which loses no generality, since $a$ is a scale parameter) its cf can be written in terms of cosine and sine integrals as $${4 t^2}\psi(t)={e^{-i t} \left(2 \left(1+e^{2 i t}\right) \text{Ci}(2 t)+4 e^{i t} \text{Si}(2 t) \sin (t)+\log \left(-\frac{i}{t}\right) \\ -\log \left(-\frac{2 i}{t}\right)+\log (2 i t)-\log (4 i t)-2 (\log (t)+\gamma )-2 e^{2 i t} (\log (t)+\gamma +\log (2))\right)}$$ From that you can extract the moments.
– whuber
Dec 9, 2016 at 17:08

Given: $$X \sim \text{Triangular}(-a,m,a)$$ with pdf $$f(x)$$:

where parameter $$m$$, instead of being fixed, is itself a random variable. In particular, $$M \sim \text{Uniform}(-a,a)$$, with pdf $$g(m)$$:

Then, we seek the parameter mixture distribution $$E_{g }\big[\;f (x \; \big | \;M =m )\big]$$, which has pdf say $$h(x)$$:

where I am using the Expect function from the mathStatica package for Mathematica to automate the calculation. The domain of support is, of course, $$(-a,a)$$.

To illustrate, here is a plot of the pdf $$h(x)$$, as parameter $$a$$ varies:

The cdf $$P_h(X is:

Comparison to semi-circle distribution

The following diagram compares the pdf of the exact solution $$h(x)$$ to the semicircle pdf that the OP refers to. The pdf $$h(x)$$ is somewhat more peaked than the semi-circle:

Comparison to POWER semi-circle distribution

While the functional form is different, one can approximate the pdf $$h(x)$$ extremely well with a power semi-circle distribution of form:

$$\phi(x) = k \left(a^2-x^2\right)^{3/4}$$

where $$k = \frac{\Gamma \left(\frac{9}{4}\right)}{\left(\sqrt{\pi } a^{5/2}\right) \Gamma \left(\frac{7}{4}\right)}$$

The fit is so good that one can barely see any perceptible difference:

• +1. Note that the "power semi-circle distribution" is a shifted rescaled Beta$(7/4,7/4)$ distribution.
– whuber
Dec 9, 2016 at 23:26