Consider the probability mass function of the Poisson distribution given a mean $\lambda$:

\begin{equation} \mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !} \end{equation}

By assuming that $\lambda$ itself is a random variable, is it possible to transform the distribution of the random variable $Y$ into a power law? In other words, is there a pdf $f(\lambda)$ that turns $\mathbb{P}\left(Y=k\right)$ into the pmf of a power law:

\begin{align} \mathbb{P}\left(Y=k\right)=\int_0^\infty \mathbb{P}\left(Y=k|\lambda\right)f(\lambda)\mathrm{d}\lambda \end{align}

Also, does this type of manipulation have a name? I am sure this must be a common question, but I could not find typical examples that englobe this case.

Consider the probability mass function of the Poisson distribution given a mean $\lambda$:

\begin{equation} \mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !} \end{equation}

By assuming that $\lambda$ itself is a random variable, is it possible to transform the distribution of the random variable $Y$ into a power law? In other words, is there a pdf $f(\lambda)$ that turns $\mathbb{P}\left(Y=k\right)$ into the pmf of a power law:

\begin{align} \mathbb{P}\left(Y=k\right)=\int_0^\infty \mathbb{P}\left(Y=k|\lambda\right)f(\lambda)\mathrm{d}\lambda \end{align}

Also, does this type of manipulation have a name? I am sure this must be a common question, but I could not find typical examples that englobe this case.

Edit: After further readings, it seems that a Poisson process with random intensity is called a Cox process or a double stochastic Poisson process.