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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.

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    $\begingroup$ 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. $\endgroup$
    – whuber
    Dec 9, 2016 at 17:08

1 Answer 1

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


(source: tri.org.au)

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


(source: tri.org.au)

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


(source: tri.org.au)

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:


(source: tri.org.au)

The cdf $P_h(X<x)$ is:


(source: tri.org.au)

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:


(source: tri.org.au)

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:


(source: tri.org.au)

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

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