Beta-Prime与Gamma随机变量比值的数学期望求解

First, let's state a critical default assumption here: we assume random variables $V$ and $U$ are independent. Without independence, we can't compute this expectation using only their marginal distributions.

Step 1: Recall key properties of the involved distributions #

• Beta-Prime Distribution ($\beta'(P, PM)$): For a random variable $V \sim \beta'(a, b)$, the expectation exists if and only if $b > 1$, and is given by:
  $$E[V] = \frac{a}{b - 1}$$
  Substituting $a = P$, $b = PM$, this simplifies to:
  $$E[V] = \frac{P}{PM - 1}$$
  This is valid only when $PM > 1$ (i.e., $M > \frac{1}{P}$).

• Gamma Distribution ($\Gamma(PM, \zeta)$): Gamma distributions are commonly parameterized in two ways—we'll cover both cases since the problem doesn't specify:

  1. Rate parameterization: $\Gamma(k, \lambda)$ where $k$ is shape, $\lambda$ is rate. The expectation of $1/U$ exists if $k > 1$, and is:
     $$E\left[\frac{1}{U}\right] = \frac{\lambda}{k - 1}$$
     Substituting $k = PM$, $\lambda = \zeta$, we get:
     $$E\left[\frac{1}{U}\right] = \frac{\zeta}{PM - 1}$$
  2. Scale parameterization: $\Gamma(k, \theta)$ where $k$ is shape, $\theta$ is scale. The expectation of $1/U$ exists if $k > 1$, and is:
     $$E\left[\frac{1}{U}\right] = \frac{1}{\theta(k - 1)}$$
     Substituting $k = PM$, $\theta = \zeta$, we get:
     $$E\left[\frac{1}{U}\right] = \frac{1}{\zeta(PM - 1)}$$

Step 2: Compute the joint expectation #

Since $V$ and $U$ are independent, the expectation of their ratio splits into the product of their expectations (specifically, $E[V \cdot 1/U] = E[V] \cdot E[1/U]$):

Case 1: Gamma distribution uses rate parameterization

$$E\left[\frac{V}{U}\right] = \frac{P}{PM - 1} \cdot \frac{\zeta}{PM - 1} = \frac{P\zeta}{(PM - 1)^2}$$
Valid when $PM > 1$.

Case 2: Gamma distribution uses scale parameterization

$$E\left[\frac{V}{U}\right] = \frac{P}{PM - 1} \cdot \frac{1}{\zeta(PM - 1)} = \frac{P}{\zeta(PM - 1)^2}$$
Valid when $PM > 1$.

Alternative derivation using Gamma ratio representation of Beta-Prime #

The Beta-Prime distribution $\beta'(P, PM)$ can be written as the ratio of two independent Gamma variables: $V = \frac{X}{Y}$, where $X \sim \Gamma(P, 1)$ and $Y \sim \Gamma(PM, 1)$. Using independence of $X, Y, U$:
$$E\left[\frac{V}{U}\right] = E\left[\frac{X}{Y \cdot U}\right] = E[X] \cdot E\left[\frac{1}{Y}\right] \cdot E\left[\frac{1}{U}\right]$$
We know $E[X] = P$, $E[1/Y] = \frac{1}{PM - 1}$ (from Gamma rate parameterization with $\lambda=1$). Multiplying these with our earlier $E[1/U]$ results gives the same expressions as above.

内容的提问来源于stack exchange，提问作者Felipe Augusto de Figueiredo