# Probability distribution function expressed in terms of a divergent series

I'm interested in finding the CDF and PDF of $$U_i$$ defined as follows, $$U_i=\frac g{d^{\alpha}}$$ where $$g$$ is a gamma distributed random variable with shape $$k$$ and scale $$\theta$$, and $$d$$ is a random variable with distribution $$f_d(x)=\frac {2x}{R^2}, 0\le x \le R$$.
To find the CDF of $$U_i$$ $$F_{U_i}(y)=P(U_i where $$F_g(y)$$ is the CDF of gamma distributed random variable.
After performing the integration, which requires the use of the expansion of incomplete gamma function, I came to the following CDF: $$F_{U_i}(y)=\sum_{n=0}^\infty \frac {2(-1)^nR^{k\alpha+n\alpha}y^{k+n}}{\Gamma(k)n!\theta^{k+n}(k+n)(k\alpha+n\alpha+2)}$$ I want the CDF in this format because my ultimate goal is to find the CDF of $$U=\sum_{i=1}^NU_i$$ which requires me to find the Laplace transform of $$F_{U_i}(y)$$ and raise it to power $$N$$.
Now it's easy to find the PDF of $$U_i$$, i.e. $$f_{U_i}(y)=\frac {d}{dy}F_{U_i}(y)$$.
I'm looking for someone to clarify these points to me:

• My derived $$f_{U_i}(y)$$ doesn't make sense to me, because it's based on a divergent series. How can it's integration from zero to infinity equals one?
• In Matlab I took the $$K^{th}$$ elements of the series, and plotted it along with the histogram of 1000 $$U_i$$'s, the two plots are not related.
• If you look at the definition of $$U_i$$, $$U_i$$ can take values from 0 to infinity, high values take less probability. This fact is not clear in the derived distribution $$f_{U_i}(y)$$. What is the domain of the derived PDF? What is the integration limit if I wannna find $$F_\epsilon(\epsilon)=\int f(y)F_{U_i}(\epsilon(y+\sigma^2))dy$$? where $$f(y)$$ is another density function.
• How can I simulate the derived PDF in Matlab to check if it's really correlated with $$U_i$$
• Please explain why you think this series diverges. As far as I can tell, for most values of $\alpha$ (including all positive values) it is an entire function: it converges everywhere in the complex plane.
– whuber
Jan 18, 2020 at 15:12
• @whuber if it converges how do you proof $\int_0^\infty f_{U_i}(y) dy=1$? Jan 18, 2020 at 20:04
• That's irrelevant, because a more basic problem is that $F_{U_i}$ obviously is not a CDF, since it is directly proportional to $\theta^{-k}.$ Thus, there is at most one $\theta$ that could possibly make this a valid CDF and for arbitrary $\theta$ and $k$ it cannot be a CDF.
– whuber
Jan 18, 2020 at 20:18
• If it's not a CDF then was my derivation wrong? Jan 18, 2020 at 20:29
• It must have been. There are some simple checks. One of them is that because $\theta$ is a scale parameter for $g,$ it must be a scale parameter for $U_i,$ whence the CDF must be a function of $y/\theta$--but it is not. Another is to plug in simple values of parameters. E.g., $\alpha=1,$ $k=1,$ $R=1$ gives a sum that evaluates to $(y^2 + 2e^{-y}(y+1)-2)/y^2,$ which works, so you might be close.
– whuber
Jan 18, 2020 at 20:38

The series converges to a distribution function. It can be evaluated in closed form.

Upon identifying the terms varying with $$n,$$ write your function in a simpler form as

$$\begin{equation*} F_{U_i}(y)=\frac{2(R^{\alpha}y/\theta)^k}{\Gamma(k)}\sum_{n=0}^\infty \frac {(-R^\alpha y / \theta)^n}{n!(k+n)((k+n)\alpha+2)} \\ = \frac{2x^k}{\Gamma(k)}\sum_{n=0}^\infty \frac{(-x)^n}{n!\,(k+n)((k+n)\alpha+2)} \end{equation*}$$

for $$x = R^\alpha y /\theta.$$ In other words, $$\sigma=\theta/R^\alpha$$ is a scale parameter.

Assuming (from the form of the expression $$g/d^\alpha$$) that $$\alpha \gt 0,$$ every one of the terms in the sum is bounded above in size by $$x^n / n!$$ showing the sum is dominated by the absolutely convergent series for $$\exp(x),$$ whence the sum converges absolutely for all $$x.$$

To evaluate such a sum we will use partial fractions. Consider the simpler function

$$h(x, a) = \sum_{n=0}^\infty \frac{(-1)^n x^n}{n! (a+n)}$$

with $$a \ge 0.$$ Similar considerations demonstrate absolute convergence so we may differentiate term by term to obtain

$$\frac{\mathrm{d}}{\mathrm{d}x} \left(x^a h(x,a)\right) = \sum_{n=0}^\infty \frac{(-1)^n x^{n+a-1}}{n!} = x^{a-1}\sum_{n=0}^\infty \frac{(-1)^n x^{n}}{n!} = x^{a-1} e^{-x}.$$

Therefore

$$h(x,a) = x^{-a} \int^x t^{a-1} e^{-t}\mathrm{d}t = C + x^{-a}\,\gamma(a, x),$$

where $$\gamma$$ is the lower Incomplete Gamma Function and $$C$$ is a constant of integration. It can be found by applying L'Hopital's Rule to $$x^{-a}\int_0^x t^{a-1} e^{-t}\mathrm{d}t$$ as $$x\to 0^+$$ and comparing the resulting limit of $$C+1/a$$ to $$h(0,a) = (-0)^0/(0!(a+0)) = 1/a$$ to conclude $$C=0.$$

Now because

$$\frac{1}{(k+n)((k+n)\alpha+2)} = \frac{1}{\alpha}\frac{1}{(k+n)(k+n+2/\alpha)} = \frac{1}{2}\left(\frac{1}{k+n} - \frac{1}{k+n+2/\alpha}\right),$$

we again exploit the absolute convergence of the series for $$F_{U_i}$$ and set $$a=k,$$ $$b=k+2/\alpha$$ to express it as

$$F_{U_i}(y) = \frac{2x^k}{\Gamma(k)}\sum_{n=0}^\infty \frac{(-x)^n}{n!\,(k+n)((k+n)\alpha+2)} = \frac{\gamma(k,x) - x^{-\frac{2}{\alpha}}\gamma(k+\frac{2}{\alpha},x)}{\Gamma(k)}$$ where $$x = y/\sigma = R^\alpha y / \theta \text{ and }\sigma=\theta/R^\alpha.$$

Differentiating this to obtain the density is straightforward, yielding

$$\frac{\mathrm{d}}{\mathrm{d}x} F^\prime_{U_i}(y) = \frac{2\gamma(k+2/\alpha, x)}{\alpha \Gamma(k) x^{1 + 2/\alpha}} \ge 0$$

and it is also elementary to establish that

$$\lim_{y\to\infty} F_{U_i}(y) = 1 \text{ and } \lim_{y\to 0^+} F_{U_i}(y) = 0.$$

Therefore $$F_{U_i}$$ is the CDF of a continuous random variable.

Almost any statistical computing platform will compute the CDF of Gamma variables (or, equivalently, of Chi-squared variables). This is the normalized version of $$\gamma.$$ For instance, here are R implementations of $$F_{U_i}$$ and its derivative:

# CDF
pFU <- function(x, k, alpha, scale=1) {
x <- x / scale
h <- function(x, a)
ifelse(x > 0, exp(pgamma(x, a, log.p=TRUE) + lgamma(a)), 1/a)
(h(x, k) - x^(-2/alpha) * h(x, k + 2/alpha)) / gamma(k)
}

# PDF
dFU <- function(x, k, alpha, scale=1) {
x <- x / scale
ifelse(x <= 0, 0,
2/alpha * exp(-(1 + 2/alpha)*log(x) + pgamma(x, k+2/alpha, log.p=TRUE) +
lgamma(k+2/alpha) - lgamma(k))) / scale
}


Here are plots of them using the curve function:

The underlying red curves use these functions. As a check, overplotted in black are direct implementations of the original series (for the CDF) and a numerical derivative of pFU (for the PDF).

• Great!!! Thank you very much. Jan 21, 2020 at 2:50
• How do you integrate the derived pdf from 0 to infinity and you get 1? How can I calculate the expected value? Feb 2, 2020 at 20:31
• (1) $F_{U_i}$ already is the integral of the PDF, so just take its limit as the argument goes to $\infty.$ (2) Compute the expected value by integrating $1-F_{U_i}(x)$ from $0$ to $\infty.$ Because $F_{U_i}$ has been expressed in terms of $\gamma,$ this looks is fairly simple. I would integrate by parts.
– whuber
Feb 2, 2020 at 21:21
• please have a look at my new question. I need $E[x^2]$also Feb 2, 2020 at 21:26