Is there a simpler probabilistic causal model that describes this data generating process? Consider the following data generating process:


*

*A person with gender male or female is selected from a population with probability $\alpha$ of selecting female.

*The person is offered a drug to treat an illness.

*If the person is male, he decides to take the drug with probability $\beta$. If the person is female, she decides to take the drug with probability $\gamma$.    


Below I specify what I believe is a faithful probabilistic causal model (PCM) for this process, I'd like to know if there is a simpler one.  In particular:
Is there a PCM that faithfully represents the data generating process above but with fewer exogenous variables?
I don't think there is, but for some reason, I feel like the model below is needlessly complex.

Let $U_G$, $U_M$, and $U_F$ be random variables with values in $\{0,1\}$ such that 
\begin{align}
  U_G  \sim \mathrm{Bernoulli}(\alpha), \qquad
  U_M  \sim \mathrm{Bernoulli}(\beta), \qquad
  U_F  \sim \mathrm{Bernoulli}(\gamma).
\end{align}
Let functions $f_G$ and $f_D$ be defined as follows:
\begin{align}
  f_G(u_G) = u_G, \qquad f_D(g, u_F, u_M) = gu_F + (1-g)u_M,
\end{align}
and define the random variables $G$ and $D$ as
\begin{align}
  G = f_G(U_G), \qquad D = f_D(G, U_F, U_M).
\end{align}
Finally let $U = \{U_G, U_M, U_F\}$, $V = \{G, D\}$, $F = \{f_G, f_D\}$, and $P$ be the probability distribution over $U$ specified above via independent Bernoulli distributions, then $(U, V, F, P)$ is a PCM, and I believe it faithfully represents the process describes above.  

The corresponding causal graph is:

The idea here is that the exogenous variable $U_G$ captures the randomness inherent in selecting gender $G$ as either male or female with value $G = 1$ corresponding to female and value $G = 0$ corresponding to male.  The exogenous variables $U_F$ and $U_M$ capture the gender-dependent randomness in whether a person will or will not take the drug with $D = 1$ corresponding to taking the drug and $D = 0$ corresponding to not taking the drug.  The function $f_D$ is constructed to ensure that if the person selected turns out to be female, the probability of taking the drug follows the distribution of $U_M$ while if the person selected turns out to be make, the probability of taking the drug follows the distribution of $U_F$.
 A: 
Is there a PCM that faithfully represents the data generating process
  above but with fewer exogenous variables?

I am not sure I understand what you want, but if the goal is just to reduce the number of exogenous variables, you can define just one exogenous random variable $U_{D}$, instead of the two you currently have $U_{M}$ and $U_{F}$. 
Let $U_D$ and $U_{g}$ be uniformly distributed over (0,1). You can  write structural equations for your model in the form of,
$$
G = I(U_{G} > \alpha)\\
D = G\times I(U_{D} > \gamma) + (1-G)\times I(U_{D} > \beta)
$$
Where $I(\cdot)$ is the indicator function.
A: I think there should be only two random variables $G$ and $D$. $G$ maps into the set $\{M, F\}$ and $D$ maps into the set $\{0,1\}$ (0=patient did not take the drug, 1=patient took the drug). Then we define/assume ($p$ always represents densities, i.e. since all the variables are discrete, $p(d) = P[D=d], p(d,g) = P[D=d ~\text{and}~ G=g]$ and so forth)
$$p(d,g) = p(d|g)p(g)$$
and $p(d|g)$ represents the decision given the gender, i.e.
$$p(d|g) = \begin{cases} \text{Bernoulli}(\beta)(d) & \text{if $g=M$} \\ \text{Bernoulli}(\gamma)(d) & \text{if $g=F$} \end{cases}$$
in other words
$$p(d|g) = \mathbf{1}_{g=M} (\beta^d \cdot (1-\beta)^{1-d}) + \mathbf{1}_{g=F}(\gamma^d \cdot (1-\gamma)^{1-d})$$
and
$$p(g) = \alpha^{\mathbf{1}_{g=F}}(1-\alpha)^{\mathbf{1}_{g=M}}$$
or rather
$$p(d,g) = p(d|g) p(g) = \left[ \mathbf{1}_{g=M} (\beta^d \cdot (1-\beta)^{1-d}) + \mathbf{1}_{g=F}(\gamma^d \cdot (1-\gamma)^{1-d}) \right] \left( \alpha^{\mathbf{1}_{g=F}}(1-\alpha)^{\mathbf{1}_{g=M}} \right)$$
