# Sampling with fixed probability from two different distributions. How is the sample distributed?

Let

• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$\mu$$ be a probability measure on $$(\mathbb R,\mathcal B(\mathbb R))$$
• $$X$$ be real-valued random variable on $$(\Omega,\mathcal A,\operatorname P)$$
• $$\kappa$$ be a Markov kernel on $$(\mathbb R,\mathcal B(\mathbb R))$$
• $$p\in[0,1]$$

Assume we construct a real-valued random variable $$Y$$ on $$(\Omega,\mathcal A,\operatorname P)$$ in the following way: With probability $$p$$ we draw $$Y$$ from $$\mu$$ and with probability $$1-p$$ we draw $$Y$$ from $$\kappa(X,\;\cdot\;)$$.

What's the conditional distributon $$\operatorname P\left[Y\in\;\cdot\;\mid X\right]$$ of $$Y$$ given $$X$$? In particular, I want to determine the Markov kernel $$Q$$ on $$(\mathbb R,\mathcal B(\mathbb R))$$ such that $$\operatorname P\left[Y\in B\mid X\right]=Q(X,B)\;\;\;\text{almost surely for all }B\in\mathcal B(\mathbb R).\tag1$$

In order to give a rigorous answer, I think that we need to introduce a $$\{0,1\}$$-valued $$p$$-Bernoulli distributed random variable $$Z$$ on $$(\Omega,\mathcal A,\operatorname P)$$ such that

1. $$X$$ and $$Z$$ are independent
2. $$X$$ and $$Y$$ are independent given $$\{Z=1\}$$
3. $$\operatorname P\left[Y\in B\mid Z=1\right]=\mu(B)$$ for all $$B\in\mathcal B(\mathbb R)$$
4. $$\operatorname P\left[Y\in B\mid X\right]=\kappa(X,B)$$ almost surely on $$\{Z=0\}$$ for all $$B\in\mathcal B(\mathbb R)$$

At first glance, I thought this would be an easy task. However, I don't know how I need to proceed. First of all, is my (supposed to be equivalent) description of the problem with the random variable $$Z$$ correct or did I impose any false assumption?

If the description is correct, how do we need to proceed?

Please take note of this related question: https://math.stackexchange.com/q/3305603/47771.