Since $x$ is a count variable, just state it in the assumptions. Then everything is hunky dory (whatever hunky dory is):

Assuming[{\[Lambda] > 0, \[Kappa] > 0, 0 < \[Theta] < 1, x >= 0, x \[Element] Integers},
         PDF[aDist, x]]

which evaluates to:

\[Kappa]^\[Kappa] (\[Theta] \[Lambda])^x (\[Kappa] + \[Theta]
\[Lambda])^(-x - \[Kappa]) Binomial[-1 + x + \[Kappa], -1 + \[Kappa]]

Less heathen, past self, I think you'll agree.

Bottom line: always state all of your assumptions to guarantee best chance of simple expressions.

Since $x$ is a count variable, just state it in the assumptions. Then everything is hunky dory (whatever hunky dory is):

Assuming[{λ > 0, κ > 0, 0 < θ < 1, x >= 0, x ∈ Integers},
         PDF[aDist, x]]

which evaluates to:

κ^κ (θ λ)^x (κ + θ λ)^(-x - κ) Binomial[-1 + x + κ, -1 + κ]

Less heathen, past self, I think you'll agree.

Bottom line: always state all of your assumptions to guarantee best chance of simple expressions.